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27745 is a number whose difference with its Gray Code equivalent (23121) is a square, here 68^2. It forms a pair of such numbers with 27744.
27744 is a number whose difference with its Gray Code equivalent (23120) is a square, here 68^2. It forms a pair of such numbers with 27745.
27743 is a member of OEIS A278869: Sophie Germain primes p such that p+6 and p-6 are primes.
27742 is a number such that the digit 2 occurs in the first and last positions and the sum of digits is 22, here 22 even appears in one of the divisors (22).
27741 is a member of OEIS A097930: numbers in base 10 that are palindromic in bases 5 and 6, here 1341431 and 332233. The sequence begins:
27740 is gapful number (because a concatenation of its first and last digits (20) divides the number) but 20 is also equal to its sum of digits. This number marks the start of a run of three consecutive numbers (27740, 27741 and 27742) with this property. See blog post Gapful Numbers Revisited.
27739 is a member of OEIS A078851: initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6, here 4, 6, 2.
27738 is a member OEIS A235109: averages q of twin prime pairs, s.t. q concatenated to q is also average of a twin prime pair.
27737 is a 4k+1 prime and can therefore be expressed as a sum of two squares in one way only viz. 29^2+164^2.
27736 is number with an odd number of digits (>=3) whose SOD to left and right of middle digit are the same and with middle digit is equal to the arithmetic digital root of number, here 7. See Bespoken for Sequences entry.
27735 is a member of OEIS A046034: numbers whose digits are all primes. In base 6, all its digits are priime as well: 332223.
27734 is a number whose arithmetic (5) and multiplicative (8) digital roots are not digits of then number itself but whose arithmetic root is equal to the number of digits (5) in the number. The next such number is 27743 (a permutation of the digits of 27734).
27733 is a 4k+1 prime and thus it can be expressed as a sum of two squares in one way only viz. 142^2 + 87^2.
27732 (and the next number 27733) have the property that all their digits are prime and so they are members of OEIS A046034.
27731 is a member of OEIS A000124: central polygonal numbers (Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts, here n=235. In short, 27731 is a pancake number.
27730 is a number n such that n plus digit sum of n and (n+1) plus digit sum of (n+1) are both prime. Here :
27729 is a member of OEIS A053061: a(n) is the decimal concatenation of n and n^2 where n=27.
27728 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 152^2 + 68^2.
27727 is a member of OEIS A344344: starts of runs of FOUR consecutive Gray-code Niven numbers (A344341). The Gray Codes are as follows:
27726 is a member of OEIS A071927: barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all abundant numbers m<n.
27725 is a product of a 4k+1 prime squared and a 4k+1 prime and so it can be expressed as a sum of two squares in three different ways, namely 13^2+166^2, 34^2+163^2 and 110^2+125^2.
27724 is a member of OEIS A173052 partial sums of A072857 (primeval numbers: numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits).
27723 is a member of OEIS A225535: numbers whose cubed digits sum to a cube, and have more than one nonzero digit, here sum is 729 = 9^3. The sequence can be generated as follows:
27722 is a member of OEIS A036689: product of a prime and the previous number, here 167 and 166. The sequence can be generated as follows:
27721 is a member of OEIS A137199: a(n)=a(n-1)+3a(n-2)+a(n-3) where a(0)=a(1)=a(2)=1. The initial terms are:
27720 is a member of OEIS A002182: highly composite numbers: numbers n where d(n), the number of divisors of n increases to a record, here 96. See blog post
27719 is a member of OEIS A343048: a(n) is the least number whose sum of digits in primorial base equals n, here primorial base is 11 : 10 : 6 : 4 : 2 : 1 and n=34. See blog post Primorial Number Base Revisited.
27718 is a member of OEIS A333703: numbers k such that k divides the sum of digits in primorial base of all numbers from 1 to k, here 471206/27718 = 17. The sequence can be generated as follows:
27717 is a so-called Lucky Cube, meaning it is a number whose cubes contain the digit sequence “888”, here 27717^3 = 21293088810813. The numbers that satisfy from 27717 to 40000 are:
27716 has factors of 2 (raised to the 2nd power), the 4k+1 prime (13) (raised to the second power) and the 4k+1 prime 41. It can therefore be expressed as a sum of two squares in three different ways viz. 46^2+160^2, 80^2+146^2 and 104^2+130^2.
27715 is a number n such that n plus digit sum of n = 27737 and (n+1) plus digit sum of (n+1) = 27739 are both prime. See Bespoken for Sequences entry.
27714 is a member of OEIS A351382: products of four distinct primes between sphenic numbers (products of 3 distinct primes).
27713 is a sphenic number whose three distinct prime factors have no digital root in common and whose digital root is different from the digital root of the original number. See Bespoken for Sequences entry.
27712 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 96^2 + 136^2.
27711 is a number that reaches the palindrome 999999 after five steps of the reverse and add algorithm.
27710 is the third member of an interesting number chain (which is base independent):
27709 is a product of a 4k+3 prime (11) raised to an even power (2) and a 4k+1 prime (229). Thus it can be expressed as a sum of two squares in one way only viz. 22^2 + 165^2.
27708 is equal to the sum of 5^5 + 6^5 + 7^5.
27707 is a composite numbers such that the sum of its proper divisors is a palindrome, here 373 and also prime. In the range from this number up to 40000, there are only 26 numbers with a sum of proper divisors that is both palindromic and prime. These are (permalink):
27706 is a member of OEIS A309439: number of prime parts in the partitions of n into 10 parts, here n=43. This can be confirmed as follows:
27705 is a member of OEIS A319742: numbers k such that 345*2^k+1 is a Proth prime. See blog post Proth Numbers from January 2020.
27704 is a member of OEIS A171639: sum of n-th nonprime number and n-th noncomposite number, here n = 2722 and 27704 = 3171 + 24533. See Bespoken for Sequences entry.
27703 is a member of OEIS A083625: starting positions of strings of three 6's in the decimal expansion of Pi. See my blog post titled The Digits of Pi from July 2016.
27702 is a member of OEIS A228964: smallest sets of seven consecutive abundant numbers in arithmetic progression (the initial abundant number is listed). Here the sequence is 27702, 27708, 27714, 27720, 27726, 27732, 27738.
27701 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 70^2 + 151^2.
27700 is a product of a power of 2, a 4k+1 prime raised to the power 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in three different ways viz 12^2+166^2 and 58^2+156^2 and 90^2+140^2.
27699 is a composite numbers such that the sum of its proper divisors is a palindrome, here 14541. In the range up to 40000, the percentage of numbers withh this property is 1.89%. The numbers are:
27698 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal. See blog post titled Odds and Evens: Statistics. How many "captives" are captured by the "attractor" 27698? The answer is 7 and the captives are 27677,27679,27683,27690,27692,27694,27696. Permalink.
27697 is a 4k+1 prime and can therefore be expressed as a sum of two squares in one way only viz. 111^2+124^2.
27696 is a member of OEIS A100329: a(n) = -a(n-1) -a(n-2) -a(n-3) +a(n-4), a(0)=0, a(1)=1, a(2)=-1, a(3)=0. The sequence can be generated as follows (permalink):
27695 is a sphenic number that is the product of primes that are the smaller of twin prime pairs, here 7, 31 and 193. See blog post Triple Strength Sphenic Numbers And More.
27694 is a sphenic number arising from OEIS A181622: sequence starting with 1 such that the sum of any two distinct terms has three distinct prime factors. This sequence begins:
27693 is a product of an even power of a 4k+3 prime and two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 78^2+147^2 = 93^2+138^2.
27692 is a so-called Lucky Cube, meaning it is a number whose cubes contain the digit sequence “888”, here 27692^3 = 21235523357888. See Bespoken for Sequences entry.
27691 is a member of OEIS A174402: primes such that applying "reverse and add" twice produces two more primes, here 47363 and 83737. The sequence leading to the palindrome is (permalink): [27691, 47363, 83737, 157475, 732226, 1354463, 4998994]
27690 is a member of OEIS A376380: products of FIVE distinct primes that are sandwiched between twin prime numbers. The sequence can be generated as follows:
27689 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 133^2 + 100^2.
27688 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 38^2 + 162^2.
27687 is a member of OEIS A351866: numbers n such that sigma(n) = tau(n)! where sigma(n) is the sum of divisors and tau(n) is the number of divisors.
27686 is a composite number that has no digits in common with its arithmetic derivatives, here 14315. See Bespoken for Sequences entry.
27685 is a product of a 4k+3 prime raised to an even power and two 4k+1 primes. Thus it can be expressed as a sum of two squares in two ways viz. 42 ^2 + 161^2 and 63^2 + 154^2.
27684 is a product of a power of 2, a 4k+3 prime raised to an even power and a 4k+1 prime. Thus it can be expressed as a sum of two squares in one way only viz. 72^2 + 150^2.
27683 is a sphenic numbers in which all three primes are weak since 19 is closer to 17 than 23, 31 is closer to 29 than 37 and 47 is closer to 43 than 53. See blog post Triple Strength Sphenic Numbers And More.
27682 is product of 2 and 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 49^2 + 159^2.
27681 is a member of OEIS A319738: numbers whose Collatz trajectories cross their initial values a record number of times, here 61 times. The records are as follows (permalink):
27680 is a product of a power of 2 and two 4k+1 primes and so it can be expressed a sum of two squares in two different ways viz. 28^2+164^2 and 76^2+148^2.
27679 is a semiprime n such that n-2 is also a semiprime and both have prime digit sums and prime sum of proper divisors.
27678 is a member of OEIS A097546: denominators of "Farey fraction" approximations to Pi. The fraction is 86953/27678 = 3.14159260062143 ... with 3.14159265358979 ... being the value of pi. See blog post Farey Fractions.
27677 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 11^2 + 166^2 and 74^2 + 149^2.
27676 is a member of OEIS A379264: pentagonal numbers of the form k*(3*k-1)/2) that are abundant. Of the 193 pentagonal numbers between 1 and 40000, 55 are abundant. 27676 is the 136-th pentagonal number.
27675 is a member of OEIS A098743: number of partitions of n into aliquant parts (parts that do not divide n). This can be confirmed using the following algorithm:
27674 is a product of 2 and two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 55^2+157^2 and 85^2+143^2.
27673 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 112^2+123^2.
27672 is a member of OEIS A061317: split positive integers into extending even groups and sum: 1+2, 3+ ... +6, 7+ ... +12, 13+ ... +20, ... The sequence can be generated as follows:
27671 is a member of OEIS A245475: numbers n such that the sum of digits (23), sum of squares of digits (139), and sum of cubes of digits (911) are all prime. It is a permutation of the lowest possible number (12677) formed from the digits 1, 2, 6, 7 and 7.
27670 is a member of OEIS A022096: Fibonacci sequence beginning 1, 6. The sequence begins:
27669 is a member of OEIS A179250: numbers with 10 terms in their Zeckendorf representation, here:
27668 is product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two square in one way only viz. 52^2 + 158^2.
27667 is a member of OEIS A331846: number of compositions (ordered partitions) of n into distinct squarefree parts, here n=38. Confirmation of the number for n=38 can be obtained as follows:
27666 is a product of 2, a 4k+3 prime raised to an even power and two 4k+1 primes. Thus it can be expressed as a sum of two squares in two different ways viz. 21^2 + 165^2 and 105^2 + 129^2.
27665 is a number n such that n plus digit sum of n and n-1 plus digit sum of n-1 are both prime, here 27665 + 26 = 27691 (prime) and 27664 + 25 = 27689 (prime).
27664 is a member of OEIS A138129: multiples of 1729, the Hardy-Ramanujan number, here 16 * 1729. The sequence begins:
27663 is the END of a run of five consecutive semiprimes with only one non-semiprime intervening. Here the semiprimes are:
27662 is a numbers whose arithmetic and multiplicative digital roots are not digits of the number itself but whose arithmetic root is equal to the number of digits in the number. Here the arithmetic digital root is 5 and the multiplicative digital root is 0. The number of digits in the number (5) corresponds to the arithmetic digital root.
27661 is a semiprime whose average of its prime factors is a perfect power, here 169 = 13^2. The remaining such numbers up to 40000 are:
27660 is a member of OEIS A102623: number of compositions iof n nto a prime number of distinct parts, here n=31.
27659 is a semiprime that contains 7 as a digit of the number itself and also of both factors.
27658 is a product of a 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 163^2 + 33^2.
27657 is a member of OEIS A182279: numbers n such that 30n + {11, 13, 17, 19, 23} are five consecutive primes. The sequence can be generated as follows:
27656 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 10^2+166^2.
27655 is a composite number containing the digit 5 at least once whose prime factors each contain the digit 5 as well so that, overall, the digit 5 occurs five times. See blog post A Multiplicity of Digits: Part 1. There are only 27 such numbers in the range up to 40000:
27654 is a member of OEIS A048367: a(n)^3 is smallest cube containing exactly n 2's, here 27654^3 = 21148222722264 with seven 2's. The sequence can be generated as follows (the algorithm is flexible and any single digit can replace the 2 in the variable values of "number"):
27653 is a 4k+1 prime and can thus be expressed as a sum of two squares in one way only viz. 113^2+122^2,
27652 is a number such that the digit 2 occurs in the first and last positions and the sum of digits is 22, here 22 even appears in one of the divisors (223). This is the only number with this property in the range between 1 and 40000.
27651 is a member of OEIS A229545: numbers n such that n + (sum of digits of n) is a palindrome, here 27651 + 21 = 27672. See blog post Hidden Palindromes. Members from 27651 to 40000 are:
27650 is the sum of three consecutive integers: 95^2 + 96^2 + 97^2.
27649 is a member of OEIS A062670: composite and every divisor (except 1) contains the digit 4. There are 25 such numbers in the range up to 40000. For the digits 6 and 8 there are only seven. For digits 1, 2, 3, 5, 7 and 9, there are many.
27648 is a member of OEIS A105779: numbers n such that n + (sum of prime factors of n) = next prime after n. Here 27648 + 5 = 27653.
27647 is a member of OEIS A272285: primes of the form 43*n^2 - 537*n + 2971 in order of increasing nonnegative values of n, here n=31. The sequence up to n=50 can be generated as follows:
27646 is a sphenic numbers whose sum of prime factors is a palindrome, here 626.
27645 is a member of OEIS: numbers with odd number of digits (>=3) whose SOD to left and right of middle digit are the same and whose middle digit is equal to the arithmetic digital root, here 9.
27644 is a member of OEIS A067796: numbers k such that euler_phi(k) + euler_phi(k+1) = k, here 27644 --> 13820
27643 is a member of OEIS A064125: numbers k such that k and k+1 have the same sum of unitary divisors, here 34560. The sequence can be generated as follows:
27642 is a member of OEIS A319544: a(n) = 1*2*3*4 - 5*6*7*8 + 9*10*11*12 - 13*14*15*16 + ... - (up to n). The sequence can be generated as followed (I've to show absolute values rather than the positive/negative values shown in the OEIS entry.
27641 is a member of OEIS A108540: Golden semiprimes: a(n)=p*q and abs(p*phi-q)<1, where phi = golden ratio = (1+sqrt(5))/2. See blog post Golden Semiprimes and More About Golden Semiprimes.
27640 is member of OEIS A225534: numbers whose sum of cubed digits is prime, here 631.
27639 is a member of OEIS A156954: integers N such that by insertion of + or - or * or / or ^ between each of its digits, without any grouping parentheses, the original number N is returned. In this case, the arrangement is 2^7*6^3-9.
27638 is a sphenic number whose three distinct prime factors have no digital root in common and whose digital root is different from the digital root of the original number. The sequence can be generated as follows (permalink):
27637 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two ways viz. 9^2 + 166^2 and 114^+ 121^2.
27636 is a number such that the sums of prime and non-prime digits are equal, here 12. See blog post Prime and Non-prime Digit Sequence.
27635 is a number n that does not contain the digit 0 and that has two distinct prime factors such that n + SOD(n) and n + POD(n) are both numbers with two distinct prime factors. Here SOD(27635) = 23 and POD(27635) = 1260:
27634 is a product of 2 and two 4k+1 primes and can therefore be expressed as a sum of two squares in two different ways viz. 65^2 + 153^2 and 97^2 + 135^2.
27633 is a member of OEIS A260906: numbers n such that 3*n and n^3 have the same digit sum, here SOD(82899) = SOD(21100080445137) = 36.
27632 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal. See blog post Revisiting Odds And Evens.
27631 is a member of OEIS A225077: smaller of the two consecutive primes whose sum is a triangular number, here larger prime is 27647 and combined sum is 55278.
27630 is a member of OEIS A050789: consider the Diophantine equation x^3 + y^3 = z^3 - 1 (x < y < z) or 'Fermat near misses'. Sequence gives values of y, here x = 17328 and z = 29737.
27629 is a semiprime that contains 7 as a digit of the number itself and also of both factors. See Bespoken for Sequences entry. An algorithm to determine such semiprimes up to 40,000 is as follows:
27628 is a member of OEIS A124177: mapping f sends n to n + (sum of even digits of n) - (sum of odd digits of n) and sequence gives numbers n s.t. f^k(n) = n for some k (here 2). See blog post Revisiting Odds And Evens.
27627 is the MIDDLE of a triplet of adjacent composite numbers such that all are only one step away from their home primes, here home prime is 39209.
27626 is the smallest of a triplet of consecutive composite numbers such that all are only one step away from their home primes, here home prime is 219727.
27625 is a member of OEIS A016032: smallest integer that is sum of two integers in exactly 8 different ways. See blog post A Plethora Of Squares.
27624 is a member of OEIS A007588: stella octangula numbers: a(n) = n*(2*n^2 - 1), here n = 24. See blog post Stella Octangula.
27623 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal. See blog post titled Odds and Evens: Statistics. How many "captives" are captured by the "attractor" 27623? The answer is 3 and the captives are 27555, 27575 and 27597. Permalink
27622 is a member of OEIS A048131: becomes prime or 4 after exactly 9 iterations of f(x) = sum of prime factors of x (with multiplicity). The progression is: 27622, 1982, 993, 334, 169, 26, 15, 8, 6, 5 and 5 is prime.
27621 is a member of OEIS A028980: numbers whose sum of divisors is palindromic, here 46464.
27620 is a product of a power of 2 and two 4k+1 primes and so it can be expressed as a sum.of two squares in two different ways viz. 8^2+166^2 and 106^2+128^2.
27619 is a member of OEIS A330441: semiprimes p×q such that the concatenations of p and q in both orders are prime, here 71389 and 38971. It can be noted that 71389 is a palindromic prime since 98317 is also prime. See blog post Biprime Prime Time.
27618 is a member of OEIS A071927: barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all abundant numbers m<n.
27617 is a 4k+1 prime and can thus be expressed as a sum of two squares in one way only viz. 119^2 + 116^2.
27616 is a member of OEIS A358782: number of regions formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter, here n=21. See attached screenshot for case of n=20.
27615 is a member of OEIS A124494: numbers k for which 2*k-1, 4*k-1, 8*k-1 and 16*k-1 are primes. The initial members are:
27614 is a xenodrome that can be split into two halves such that 2^2 + 7^2 = 6^2 + 1^2 + 4^2.
27613 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 37^2 + 162^2 = 117^2 + 118^2. Note that in the latter of the two representations, the numbers are consecutive (117 and 118).
27612 is a member of OEIS A170928: least magic constant of magic squares using Smith numbers (composite numbers with sum of their digits the same as the sum of the digits of their prime factorization). Follow this Russian link.
27611 is a number that is inconsummate, self and untouchable. The previous such number was 27444 which prompted my blog post earlier in the year.
27610 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal.
27609 is a number with an odd number of digits whose sums of digits to left and right of the middle digit are the same (9) and whose middle digit is equal to the arithmetic digital root of the number, here 6.
27608 is a number with no repeating digits such that the additive digital root (here 5) is different to any of the digits of the number.
27607 is a member of OEIS A287634: Ulam numbers k such that 4*k and 16*k are also Ulam numbers. Here we have 27607 = 2 + 27605
27606 is a member of OEIS A065320: 53 'Reverse and Add' steps are needed to reach a palindrome. The palindrome reached is 4668731596684224866951378664.
27605 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two ways viz. 7^2 + 166^2 and 94^2 + 137^2.
27604 is a number with no repeating digits and whose (additive) digital root (here 1) is different to any of the digits of the number.
27603 is a number whose difference with its Gray Code equivalent (24122) is a square, here 3481 = 59^2.
27602 is a product of 2 and two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 41^2+161^2 and 91^2+139^2.
27601 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal. These numbers are what I have termed attractors. See my blog post Revisting Odds and Evens.
27600 is number whose sum of prime factors (counted with multiplicity) is a number whose digits are identical, here 44.
27599 is a number n such that n is sphenic and all three factors have at least one digit in common, here 1.
27598 is a number without the digit 0 with two distinct prime factors such that n + SOD(n) and n + POD(n) both have two distinct prime factors. Here sum of digits is 31 and product of digits is 5040 and the semiprimes are:
27597 marks the start of a run of six consecutive numbers that are only one step removed from their home primes. See blog post Record Runs Involving Home Primes.
27596 is a member of OEIS A245370: number of compositions of n into parts 3, 5 and 9, here n=54 and so an example of such a composition is 3, 3, 9, 3, 3, 9, 3, 9, 3, 9.
27595 is a member of OEIS A082550: number of nonempty subsets of {1, 2, ..., n} that contain n and have a sum that is divisible by n, here n=19. Alternatively, the number of sets of distinct positive integers whose arithmetic mean is an integer, the largest integer of the set being n.
27594 is a member of OEIS A052018: numbers k with the property that the sum of the digits (SOD) of k is a substring of k, here SOD = 27. In this case 27 also divides the number.
27593 is a product of two 4k+1 primes and so it can be expressed as the sum of two squares in two different ways viz. 32^2+163^2 and 67^2+152^2.
27592 is product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 166^2 + 6^2.
27591 is a member of OEIS A123782: number of ways to build a contiguous building with n LEGO blocks of size 1 x 4 on top of a fixed block of the same size, here n = 3. See blog post LEGO Mathematics.
27590 is a member of OEIS A263876: numbers n such that n^2 + 1 has two distinct prime divisors less than n, here 27590^2+1 = 761208101 = 53^3 * 5113.
27589 is a member of OEIS A139284: comma sequence (analog of A121805) but starting with 2. See blog post The Commas Sequence.
27588 has the property that its product of digits (4480 = 128 x 35) is a multiple of the sum of its prime divisors (35). The numbers with this property from 27588 up to 40000 are as follows:
27587 is a member of the commas sequence OEIS A121805. See my blog post The Commas Sequence. Here are the terms from 27587 to 40000:
27586 is a product of 2 and two 4k+1 primes and so it can be expressed as a sum of two squares in two ways viz. 19^2+165^2 and 81^2+145^2.
27585 is a product of a 4k+3 prime raised to an even power (2) and two 4k+1 primes. Thus is can be expressed as a sum of two squares in two different ways viz. 48^2 + 159^2 and 57^2 + 156^2.
27584 is a member of OEIS A046411: composite numbers the concatenation of whose prime factors is a prime, here 222222431. See Bespoken for Sequences entry.
27583 is a member of OEIS A296187: Yarborough primes that remain Yarborough primes when each of their digits are replaced by their squares, here 44925641. See blog post titled Yarborough and Anti-Yarborough Primes.
27582 is a member of OEIS A071927: barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all abundant numbers m<n. The initial members from 27582 upwards are
27581 is a Sophie Germain and the lesser to two twin primes.
27580 is a member of A002858: Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms. In this case we have: 27580 = 339 + 27241
27579 is a member of OEIS A301500: number of compositions of n into squarefree parts (A005117) such that no two adjacent parts are equal (Carlitz compositions).
27578 is a product of 2 and 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 107^2 + 127^2.
27577 is a member of OEIS A046034: numbers whose digits are primes. The remaining 27**** numbers are:
27576 is the sum of the 2711 th non-prime and prime numbers viz. 3157 + 24419.
27575 is a member of OEIS A173092: numbers k such that 3k-4, 3k-2, 3k+2, and 3k+4 are primes, here 82721, 82723, 82727 and 82729.
27574 is a Hidden Beast Number: in base 8 the number's representation contains three adjacent 6's (65666).
27573 is a member of OEIS A147619: numbers n = concat(a, b) such that totient(n) = totient(a) * totient(b) , here a=275 and b=73 with totient(275) = 200, totient(73)=72 and totient(27573) = 1440.
27572 is a product of a power of 2 and two 4k+1 primes, thus it can be expressed as a sum of two squares in two ways viz. 4^2+166^2 and 26^2+164^2.
27571 is a number without the digit 0 with two distinct prime factors such that n + SOD(n) and n + POD(n) both have two distinct prime factors. Here the semiprimes are:
27570 is a member of OEIS A155023: values of n such that n^a + a and n^a - a are primes where a=11. The sequence can be generated as follows (permalink):
27569 is a member of OEIS A055480: energetic number that can be represented as a sum of positive powers of its substrings, here 27^3 + 5^5 + 69^2. It forms a pair with 27568 = 2^12 + 7 + 5^6 + 6^5 + 8^2 which, additionally, is a d-powerful number.
27568 is a member of OEIS A085844: numbers equal to a permutation of the digits of the sum of their proper divisors. The sequence can be generated as follows (permalink):
27567 is a number such that the "Number Within a Number" (27) is the Sum of Digits (SoD) of the number as well as a factor of the number, its totient and the determinant of its circulant matrix. See blog post titled "More Numbers Within Numbers".
27566 is a composite number with four distinct prime factors each of which has a digit sum that is prime. The numbers with this property from 27566 up to 40000 are:
27565 is a product of three 4k+1 primes and so it can be expressed as a sum of two squares in four different ways viz. as the sum of squares of the following tuples: (3, 166), (51, 158), (54, 157) and (102, 131).
27564 is a member of OEIS A238657: number of partitions of n having standard deviation > 5. Here n=48. The sequence begins:
27563 is a number whose sum of digits about its centre point is the same, here 9. See Bespoken for Sequences entry. This number also has the property that the middle digit is equal to the arithmetic digital root of 5. There are 270 five digit numbers with this property (in the range up to 40,000).
27562 is a product of 2 and a 4k+1 prime and can therefore be expressed as a sum of two squares in one way only viz. 151^2 + 69^2.
27561 is a member of OEIS A361696: semiprimes of the form k^2 + 5, here k = 166. The sequence can be generated as follows (permalink):
27560 is a product of a power of 2 and three 4k+1 primes so that it can be expressed as a sum of two squares in four different ways viz. as the sum of squares of tuples: (2, 166), (62, 154), (86, 142), (98, 134).
27559 is a member of OEIS A046528: numbers that are a product of distinct Mersenne primes (3, 7, 31, 127, 8191 etc.). the sequence can be generated as follows (permalink):
27558 is a member of OEIS A052018: numbers k with the property that the sum of the digits of k is a substring of k. Here the sum of the digits is 27. The members of the sequence from 27558 to 40000 is as follows:
27557 is a product of two 4k+1 primes and can therefore be expressed as a sum of two squares in two different ways viz. 1^2+166^2 and 79^2+146^2.
27556 is a member of OEIS A228878: happy squares: squares whose trajectory under iteration of sum of squares of digits map includes 1, here 166^2.
27555 is a member of OEIS A341780: starts of runs of three consecutive anti-tau numbers (see OEIS A046642). See blog posts titled Anti-tau Numbers and Tau Numbers. The sequence can be generated as follows:
27554 is a number whose sum of digits about its centre point is the same, here 9. See Bespoken for Sequences entry. This number also has the property that the middle digit is equal to the arithmetic digital root of 5. There are 270 five digit numbers with this property (in the range up to 40,000). They can be found as follows (permalink):
27553 has the property that all its digits are prime and so it is a member of OEIS A046034. It forms a pair with 27552.
27552 is a member of OEIS A060678: numbers n such that sigma (x) = n has exactly 11 solutions. These solutions are: [9780, 11796, 12714, 13748, 14996, 19149, 20049, 22955, 23309, 27221, 27551].
27551 is a member of OEIS A096342: primes of the form p*q + p + q, where p and q are two successive primes, here p=163 and q=167.
27550 is a member of OEIS A256646: 26-gonal pyramidal numbers: a(n) = n*(n+1)*(8*n-7)/2, here n=19.
27549 is a member of OEIS A107085: numbers n such that in decimal representation the largest digit is equal to the digital root, here 9. See entry for 27340.
27548 is a member of OEIS A257105: composite numbers n such that n'=(n+8)', where n' is the arithmetic derivative of n, here 28220. The sequence can be generated using the following algorithm:
27547 is a member of OEIS A071320: least of four consecutive numbers which are cubefree but not squarefree. The sequence can be generated as follows (permalink):
27546 is a member of OEIS A071927: barely abundant numbers: such that sigma(n)/n < sigma(m)/m for all abundant numbers m<n.
27545 is a number whose sum of digits about its centre point is the same, here 9. See Bespoken for Sequences entry. This number also has the property that the middle digit is equal to the arithmetic digital root of 5. There are 270 five digit numbers with this property (in the range up to 40,000). They can be found as follows (permalink):
27544 is a member of OEIS A283392: integers m of form m = 3*p + 5*q = 5*r + 7*s where {p,q} and {r,s} are pairs of consecutive primes, here (3433, 3449) & (2293, 2297). The sequence can be generated as follows:
27543 is semiprime whose concatenation of prime factors (from lower to higher AND from higher to lower) is a prime, here 39181 and 91813. See Bespoken for Sequences entry.
27542 is a sphenic number 'p' whose associated sphenic brick area 'q' is also sphenic and whose associated area 'r' is such that p * q * r contains one digit with a frequency of at least 50% of all digits, here digit is 8.
27541 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 50^2 + 71^2.
27540 is a product of a power of 2, a 4k+3 prime raised to an even power and two 4k+1 primes. Therefore it can be expressed as a sum of two squares in two ways viz. 36^2+162^2 and 108^2+126^2.
27539 is a member of OEIS A056212: primes p whose period of reciprocal equals (p-1)/7. There are 30 such primes in the range up to 40000. The sequence can be generated as follows:
27538 is a product of 2, a 4k+3 prime raised to an even power and a 4k+1 prime. It can therefore be expressed as a sum of two squares in one way only viz. 147^2 + 77^2.
27537 is a member of OEIS A247317: numbers x such that the sum (266664) of all cyclic permutations of the numbers equals that of all cyclic permutations of its sum of divisors (37536) and all cyclic permutations of its Euler totient function (17952). See Sage Math bookmark.
27536 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 160^2 + 44^2.
27535 is a number n without the digit 0 with two distinct prime factors such that n + SOD(n) and n + POD(n) both have two distinct prime factors. Here we have:
27534 is a number whose representation in base 8 (65616) contains the digit 6 exactly three times such that the remaining non-6 digits add to 6. See Bespoken for Sequences entry and permalink.
27533 is a member of OEIS A046034: numbers with all digits prime (forms a pair with 27532).
27532 has the property that all its digits are prime and so it is a member of OEIS A046034. It forms a pair with 27533.
27531 is a member of OEIS A269344: magic sums of 3 X 3 semi-magic squares composed of squares of primes. See blog post Prime Squared Semi-Magic Squares.
27530 is a product of 2 and two 4k+1 primes and can therefore be expressed as a sum of two squares in two different ways viz. 31^2 + 163^2 and 73^2 + 149^2.
27529 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 148^2 + 75^2.
27528 is a number n such that the sum of digits cubed of n - 1 and n + 1 is prime (here 827 and 1213) but sum of digits cubed of n is not. See Bespoken for Sequences entry.
27527 is a prime number with all digits prime, sum of digits (23) prime, digits of sum of digits prime (2 and 3) and arithmetic digital root prime (5). See blog post What's Special About 27527?
27526 is a member of OEIS A022370: Fibonacci sequence beginning 2, 16. The sequence runs:
27525 is a number for which a home prime is unable to be ascertained. The first such number is 49. The following algorithm fails to find a home prime after 50 iterations.
27524 is a number such that its circulant matrix has a determinant that is a pronic number, here 10100 = 100 * 101.
27523 (and the previous number 27522) have the property that all their digits are prime and so they are members of OEIS A046034.
27522 is a member of OEIS A173719: sums of two successive primes s = prime(m) + prime(m+1) such that all digits of s are primes, here 13759 & 13763.
27521 is a product of three 4k+1 primes and can therefore be expressed as the sum of two squares in four different ways, namely as the sum of the squares of the following number pairs:(25, 164), (40,161), (89, 140), (95, 136).
27520 is a member of OEIS A107085: numbers n such that in decimal representation the largest digit is equal to the digital root, here 7.
27519 is a member of OEIS A256115: zeroless numbers n whose digit product squared is equal to the digit product of n^2. The sequence can be generated as follows (permalink):
27518 is a member of OEIS A114140: number of ordered sequences of coins (each of which has value 1, 2, 5, 10 or 20) which add to n, here n=20.
27517 is a member of OEIS A189072: semiprimes that are the sum of first n primes, here n = 106. The sequence can be generated as follows:
27516 is a number n with no repeating digits, whose additive & multiplicative roots differ from any of their digits & also from each other, here 3 & 0. The "seed" numbers, wiith digits in ascending order, are:
27515 is a member of OEIS A274182: semiprimes that are the sum of the first n odd primes for some n (here 105). The sequence can be generated in the following way:
27514 is a product of 2 and a 4k+1 prime and can therefore can be expressed as a sum of squares in one way only viz. 165^2 + 17^2.
27513 is a member of OEIS A176580: n^3 + largest square <= n^3), here 24^3 + 117^2. The sequence can be generated as follows (permalink):
27512 is a member of OEIS A066055: integers n > 10583 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 10583. The initial members are:
27511 is a sphenic number such that the digit 1 appears in the number itself as well as its three prime factors. See blog post Some Special Sphenic Numbers. The numbers with this property, up to 40000, are:
27510 is a member of OEIS A046403: numbers with exactly 5 distinct palindromic prime factors. The initial members are:
27509 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 130^2 + 103^2.
27508 is a product of 2, a 4k+3 prime (23) raised to an even power (2) and one 4k+1 prime. Therefore it can be expressed as a sum of two squares in one way only viz. 138^2 + 92^2.
27507 is sphenic number NOT containing the digit 3 but whose prime factors all contain the digit 3 and whose additive digital root is 3.
27506 is a product of 2 and two 4k+2 primes and therefore it can be expressed as a sum of two squares in two different ways viz. 59^2+155^2 and 109^2+125^2.
27505 is a product to two 4k+1 primes and so it can be expressed as a sum of two squares in two different way viz. 64^2+153^2 and 84^2+143^2.
27504 is a member of OEIS A109027: numbers with exactly 7 prime factors counted with multiplicity whose digit reversal is different & also has 7 prime factors (with multiplicity).
27503 is a member of the commas sequence. See my blog post The Commas Sequence. Here are the terms from 27503 to 40000:
27502 is a semiprime whose concatenations of factors (from lower to higher) is prime AND whose permutations of digits produce exactly three prime numbers. Here the three permutations are 5227, 52027 and 50227. The concatenated prime formed from the factors is 213751.
27501 is a member of OEIS A295689: a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 2, a(1) = 0, a(2) = 2, a(3) = 1. The sequence can be generated as follows:
27500 is a mid-millennium, untouchable, practical, abundant, Zumkeller and pseudoperfect number.
27499 is a member of OEIS A363965: binary palindromic numbers whose digit sum and aliquot sum are also binary palindromic. Here the palindromes are 110101101101011 (27499), 11111 (digit sum) and 101101101 (aliquot sum). The sequence can be generated as follows:
27498 is a member OEIS A071927: barely abundant numbers n s.t. sigma(n)/n < sigma(m)/m for all abundant numbers m<n.
27497 is a member of OEIS A337701: place two n-gons with radii 1 and 2 concentrically, forming an annular area between them. Connect all the vertices with line segments that lie entirely within that area. Then a(n) is the number of vertices in that figure. Here n=31. Attached diagram shows the case of n=5.
27496 is amember of OEIS A180226: a(n) = 4*a(n-1) + 10*a(n-2), with a(1)=0 and a(2)=1. The sequence can be generated as follows using the generating function which is x^2/(1-4*x-10*x^2) ... permalink:
27495 is a member of OEIS A349773: numbers that start a run of four consecutive triangular numbers with four distinct prime factors (these are shown below).
27494 is a composite numbers such the digits comprising its sum of factors (here 294) are all contained in the original number. See Bespoken for Sequences entry.
27493 is a semiprime whose concatenation of prime factors (from lower to higher AND from higher to lower) is a prime, here 191447 and 144719.
27492 is a Member of OEIS 287212: Ulam numbers k such that k/3 is also an Ulam number. The initial members are:
27491 is a member of OEIS A335789: a(n) = time to the nearest second at the n-th instant (n>=0) when the hour and minute hands on a clock face coincide, starting at time 0:00. See blog post Sexagesimal Numbers System.
27490 is a product of 2 and two 4k+1 primes and so it can be expresse as a sum of two squares in two different ways viz. 47^2+ 159^2 and 99^2+133^2.
27489 is a number with no repeating digits, whose additive (3) and multiplicative (0) digital roots are different from any of its digits and also from each other. See Bespoken for Sequences entry.
27488 is a composite numbers that is not solely a power of 2 such that the digit sum of each prime factor contains only the digit 2. See Bespoken for Sequences entry.
27487 is member of OEIS A113507: numbers whose square root in base 10 starts with 10 distinct digits, here 165.7920384. The algorithm to generate this sequence and the details of each numbers are as follows (permalink):
27486 is a member of OEIS A046515: numbers with multiplicative persistence value 6, here 27486, 2688, 768, 336, 54, 20, 0. There are 120 permutations of these digits, all with the same multiplicative persistence and same trajectory. These are
27485 is a number with at least THREE distinct prime factors such that the digital roots of each factor are the same, here 5. See Bespoken for Sequences entry.
27484 is a number that is the larger of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here 27483 = 3 * 9161 --> 39161 is prime and 27484 = 2^2 * 6871 --> 226871 is also prime. See Bespoken for Sequences entry.
27483 is a number that is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here 27483 = 3 * 9161 --> 39161 is prime and 27484 = 2^2 * 6871 --> 226871 is also prime. See Bespoken for Sequences entry.
27482 is a member of OEIS A297405: binary "cubes"; numbers whose binary representation consists of three consecutive identical blocks, here 11010 --> 110101101011010.
27481 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 165^2 + 16^2. The sequence can be generated as follows:
27480 is a member of OEIS A343755: number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts, here n=10. Starting from n=1, the initial members are 7, 30, 144, 474, 1324, 2934, 5797, 10614, 17424, 27480, ...
27479 is a member of OEIS A059763: primes starting a Cunningham chain of the first kind of length 4. The progression is 27479 --> 54959 --> 109919 --> 219839. All members are "unsafe" primes and all chains are exactly of length 4 and no larger.
27478 is a member of OEIS A064799: numbers that are the sum of the n-th composite and n-th prime numbers, here 27478 = 3149 + 24329 where n=2702. See Bespoken for Sequences entry.
27477 is a member of OEIS A171639: sum of the n-th non-prime and n-th prime numbers, here 27477 = 24329 + 3148 where n=2702. See Bespoken for Sequences entry.
27476 is a product of a power of 2 and a 4k+1 prime and can therefore be expressed as a sum of two squares in one way only viz. 124^2 + 110^2.
27475 is a member of OEIS A334542: numbers m s.t. m^2 = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m, here p = 1960 and k = 27405. See blog post titled Pythagorean Triangles With Integer Sides. Intitial members are
27474 is a member of OEIS A046411: composite numbers the concatenation of whose prime factors is a prime, here 2319241. See Bespoken for Sequences entry.
27473 is a numbers whose arithmetic and multiplicative digital roots are not digits of the number itself but whose arithmetic root is equal to the number of digits in the number. Here the arithmetic digital root is 5 and the multiplicative digital root is 8. The number of digits in the number (5) corresponds to the arithmetic digital root. See entry to 27374 (a permutation of the digits of 25473).
27472 is a product of a power of 2 and two 4k+1 primes, therefore it can be expressed as a sum of two squares in two different ways viz. 24^2 + 164^2 and 56^2 + 156^2.
27471 is semiprime whose concatenation of prime factors (from lower to higher AND from higher to lower) is a prime, here 39157 and 91573. See Bespoken for Sequenes entry. The nearby semiprime 27469 = 13 * 2113 has the same property
27470 has a so-called Lucky Cube because its cube contains the digit sequence “888”, here 20728886723000. See Bespoken for Sequences entry.
27469 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 30^2+163^2 and 35^2+162^2.
27468 is a member of OEIS A178213: Smith numbers of order 3. See blog post Higher Order Smith Numbers. The sequence can be generated as follows (permalink):
27467 is a member of OEIS A133539: sum of third powers of five consecutive primes, here 11, 13, 17, 19 and 23. The sequence can be generated as follows:
27466 is a member of OEIS A028980: numbers whose sum of divisors is palindromic, here 42624. See Bespoken for Sequences entry. Note that the divisors include the number itself.
27465 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal. See entry for 27326, The captives of this attractor are 27299, 27307, 27309, 27311, 27320, 27321, 27322, 27324 and 27328.
27464 is a product of a power of 2 and a 4k+1 prime number and so it can be expressed as a sum of two squares in one way only viz. 158^2 + 50^2.
27463 is a number with central digit equal to its arithmetic digital root and whose prime factors also share this root and digit. See Bespoken for Sequence entry.
27462 is a number that can be expressed as the sum of two pronic numbers in six different ways. See blog post Even Numbers as Sums and Differences of Two Pronic Numbers. the six way are:
27461 is a number n such that n plus digit sum of n (27481) and n-1 plus digit sum of n-1 (27479) are both prime. See Bespoken for Sequences entry.
27460 is a product of a power of 2 and two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 66^2+152^2 and 82^2+144^2.
27459 is a member of OEIS A052018: numbers k with the property that the sum of the digits of k is a substring of k, here the sum of the digits is 27. See entry for 27387.
27458 is a product of 2 and a 4k+1 prime and therefore it can be expressed as a sum of two squares in one way only viz. 157^2 + 53^2.
27457 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 129^2 + 104^2.
27456 is a member of OEIS A109029: numbers with exactly 9 prime factors counted with multiplicity whose digit reversal is different & also has 9 prime factors. See blog post titled Remarkable Reversals. The sequence can be generated as follows (permalink):
27455 is member of OEIS A138760: numbers n s.t. n^4 is a sum of 4th powers of four nonzero integers whose sum is n, here n = 955 + 1700 - 2364 + 5400.
27454 is a member of OEIS A034279: decimal part of a(n)^(1/4) starts with a 'nine digits' anagram, here 27454^0.25 = 12.872159346 ...The sequence can be generated as follows (permalink):
27453 is a so-called Lucky Cube number because its cube (20690425888677) contains the digit sequence “888”; From
27452 is a number that can be represented in terms of Fibonacci numbers using digits in sequence, here F(2 + F(7)) × 45 + 2. See resource: https://rgmia.org/papers/v19/v19a143.pdf and blog post Fibonacci Sequence and Selfie Numbers.
27451 is a number that can be represented in terms of Fibonacci numbers using digits in sequence, here F(2 + F(7)) × 45 + 1. See resource: https://rgmia.org/papers/v19/v19a143.pdf and blog post Fibonacci Sequence and Selfie Numbers.
27450 can expressed as a sum of two squares in three different ways viz. 15^2+165^2, 87^2+141^2 and 111^2+123^2.
27449 is a 4k+1 prime and can therefore be expressed as a sum of two squares in one way only viz. 160^2 + 43^2.
27448 is a number that is a concatenation of two cubes, here 2744 = 14^3 and 8 = 2^3 and so we have 14^3 | 2^3. See Bespoken for Sequences entry. The members of this sequence up to 40000 are:
27447 is a sphenic number whose three distinct prime factors have no digital root in common (3, 7 and 2) and all of which differ from the digital root (6) of the original number.
27446 is a semiprime with digit sum of 23 and 2 as the smaller factor. It has 23 as the last two digits of the larger factor. See blog post Human Genome Numbers. The members of the sequence up to 40,000 are:
27445 is a sphenic number whose sum of prime factors is a palindrome, here 515. See Bespoken for Sequences entry. The sequence members from 27445 up to 40000 are:
27444 is a rare number that is an inconsummate, untouchable and self number. See Bespoken for Sequences entry and blog post titled Simultaneously Inconsummate, Self and Untouchable Numbers. From 27444 to 40000, the members of the sequence are:
27443 is a number n without the digit 0 with two distinct prime factors such that n + SOD(n) and n + POD(n) both have two distinct prime factors. Here we have:
27442 is a product of 2 and a 4k+1 prime. Therefore it can be expressed as a sum of two squares in one way only viz. 161^2 + 39^2.
27441 is a product of a 4k+3 prime (3) raised to an even power (2) and a 4k+1 prime. Therefore it can be expressed as a sum of two squares in one way only viz. 135^2 + 96^2.
27440 is a number that is divisible by its first digit cubed, its second digit squared and its third digit (0's and 1's are not allowed). See Bespoken for Sequences entry. The sequence runs:
27439 is a member of OEIS A005894: centered tetrahedral numbers, (2*n + 1)*(n^2 + n + 3)/3 where n=34. Useful information about these types of numbers can be found at https://oeis.org/wiki/Centered_Platonic_numbers.
27438 is a member of OEIS A014363: aliquot sequence starting at 966 (one of the Lehmer five). The sequence runs:
27437 is a 4k+1 prime and can therefore be expressed as a sum of two squares in one way only viz. 154^2 + 61^2.
27436 is a member OEIS A143610: numbers of the form p^2 * q^3, where p,q are distinct primes. The sequence runs:
27435 is a member of OEIS A002414: octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2, here n=30. The sequence runs:
27434 is a member of OEIS A000330: square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6, here n=43. See blog post Pyramidal Numbers.
27433 is a member of OEIS A115933: numbers k such that k^3 contains a pandigital substring. The sequence can be generated as follows (permalink):
27432 is a member of OEIS A109027: numbers with exactly 7 prime factors counted with multiplicity whose digit reversal is different & also has 7 prime factors (with multiplicity). The sequence can be generated as follows (permalink):
27431 is a member of OEIS A174402: primes such that applying "reverse and add" twice produces two more primes (40903 and 71807). The sequence can be generated as follows:
27430 is a member of OEIS A107085: numbers n such that in decimal representation the largest digit is equal to the digital root, here 7; the digital root (7) is also different to that of any of its prime factors. See entry for 27340.
27429 is a member OEIS member of OEIS A020905: sum of n plus its prime factors associated with A020700 (numbers k s.t. k + sum of its prime factors = (k+1) + sum of its prime factors). The sequence can be generated as follows:
27428 is a product of a power of 2 and a 4k+1 prime. Therefore it can be expressed as a sum of two squares in one way only viz. 122^2 + 112^2.
27427 is a member of OEIS A296563: Yarborough primes that remain Yarborough primes when each of their digits are replaced by their cubes, here 8343648343. See blog post Yarborough and Anti-Yarborough Primes. The sequence can be generated as follows (permalink):
27426 is a member of OEIS A054782: number of primes <= the n-th Fibonacci number, here n=28. The sequence can be generated as follows:
27425 is.a product of a 4k+1 prime raised to the second power and a 4k+1 prime, thus it can be expressed as a sum of two squares in three different ways viz. 23^2+164^2, 68^2+151^2 and 80^2+145^2.
27424 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 32^2 + 100^2.
27423 is a member of OEIS A178919: smallest of three consecutive integers divisible respectively by three consecutive squares greater than 1 (here 9, 16 and 25). The inital members are 2223, 5823, 9423, 13023, 16623, 20223, 23823, 27423, 31023, 32975, 34623, 38223, ...
27422 is the larger of two consecutive semiprimes whose prime factors both contain three or more 1's. The sequence can be generated as follows:
27421 is a product of two 4K+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 14^2 + 165^2 and 90^2 + 139^2.
27420 is a member of OEIS A062681: numbers that are sums of 2 or more consecutive squares in more than 1 way. Here two ways, viz. 55^2 + ... + 62^2 and 4^2 + ... + 43^2. The sequence can be generated as follows (permalink):
27419 is a member of OEIS A127345: a(n) = pq + pr + qr with p = prime(n), q = prime(n+1), and r = prime(n+2). The sequence can be generated as follows (permalink):
27418 is a product of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 137^2 + 93^2.
27417 is a member of OEIS A135503: a(n) = n*(n^2 - 1)/2 when n=38 (for n > 2, a(n) is the maximum value of the magic constant in a perimeter-magic n-gon of order n.
27416 is a member of OEIS A117560: a(n) = n*(n^2 - 1)/2 - 1 where a(n-1) is an approximation for lower bound of the "antimagic constant" of an antimagic square of order n, here n=38. The initial members of the sequence are:
27415 is a semiprime whose average of prime factors is a perfect number, here (5+5483)/2 = 2744 = 14^3. See Bespoken for Sequences entry. The semiprimes from 27415 to 40000 with this property are:
27414 is a number such that adjacent numbers (27413 and 27415) both have prime sum of digits cubed (443 and 541). See blog post Sum of Digits Cubed to the Rescue. Here are a list of such numbers from 27414 up to 40000:
27413 is a number with no repeating digits and additive digital and multiplicative digital roots different from any of these digits and also from each other. Here the roots are 8 and 6 respectively. See Bespoken for Sequences entry.
27412 is a member of OEIS A074934: number of integers in {1, 2, ..., Fibonacci(n)} that are coprime to n, here n=23. The sequence can be generated as follows:
27411 is a member of OEIS A341313: a(n) = (a(n-1) + a(n-3))/2^m, where 2^m is the highest power of 2 that divides both a(n-1) and a(n-3), with a(0) = a(1) = a(2) = 1.
27410 is a product of 2 and two 4k+1 primes and therefore it can be expressed as a sum of two squares in two different ways viz. 29^2+163^2 and 113^2+121^2.
27409 is a 4k+1 prime and thus it can be expressed as a sum of two squares in one way only viz. 28^2 + 105^2.
27408 is a member of OEIS A103763: a(n) = digit reversal of A103741(n) - a(n) is a non-palindromic composite located between twin primes whose reverse, which is less than it, is also located between twin primes. The sequences can be generated as follows:
27407 is a member of OEIS A342681: primes which, when added to their reversals, produce palindromic primes, here 97879. The sequence can be generated as follows:
27406 is a sphenic number whose three distinct prime factors have no digital root in common and whose digital root is different from the digital root of the original number. See entry for 27303 and Bespoken for Sequences entry.
27405 is a member of OEIS A000332: binomial coefficient binomial(n,4) = n * (n-1) * (n-2) * (n-3)/24. The sequence runs:
27404 is a member of OEIS A096399: numbers k such that both k and k+1 are abundant. The sequence begins:
27403 is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes, here 27403 --> 67409 and 27404 = 2^2 * 13 * 17 * 31 --> 22131731. See Bespoken for Sequences entry.
27402 is a member of OEIS A071927: barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all abundant numbers m<n. See blog post Barely Abundant Numbers.
27401 is a number whose sum of prime factors (counted with multiplicity) is a number whose digits are identical, here 111. See Bespoken for Sequences entry.
27400 can be expressed as a sum of two squares in three different ways viz. 34^2+162^2, 70^2 +150^2 and 78^2+146^2.
27399 is a member of OEIS A019461: add 1, multiply by 1, add 2, multiply by 2, etc.; start with 0. The sequence can be generated as follows:
27398 is a member of OEIS A365257: the five digits of a(n) and their four successive absolute first differences are all distinct. See entry for 27391 and blog post titled Very Special Five Digit Numbers.
27397 is a 4k+1 prime and thus it can be expressed as a sum of two squares in one way only viz. 159^2 + 46^2.
27396 is a product of a power of 2, a 4k+3 prime raised to an even power and a 4k+1 prime. Thus it can be expressed as a sum of two squares in one way only viz. 120^2 + 114^2.
27395 is a zeroless number with two distinct prime factors s.t. n + SOD(n) and n + POD(n) also have two distinct prime factors where SOD = Sum of Digits and POD = Product of Digits; cyclic number. Here the resulting semiprimes are:
27394 is a product of 2 and 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 165^2 + 13^2.
27393 is a member of OEIS A034817: concatenations where 'prevprime(n) + n', 'n + nextprime(n)' and 'prevprime(n) + n + nextprime(n)' are all prime. Here the primes are:
27392 is member of OEIS A064799: sum of n-th prime number and n-th composite number, here n=2698. See Bespoken for Sequences link.
27391 is a member of OEIS A365257: the five digits of a(n) and their four successive absolute first differences are all distinct. See blog post titled Very Special Five Digit Numbers. There are 96 such numbers and they are:
27390 is a member of OEIS A046387: products of 5 distinct primes. There are 237 such numbers in the range up to 40,000. The sequence can be generated as follows (permalink):
27389 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 58^2+155^2 and 85^2+142^2.
27388 is a member of OEIS A199996: composite numbers whose multiplicative persistence is 6 (27388, 2688, 768, 336, 54, 20, 0). See blog post titled Multiplicative Persistence and Multiplicative Digital Root. Sequence members up to 40000 are:
27387 is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. 27387 gives 3317179 which is prime and 27388 = 2^2 * 41 * 167 --> 2241167 which is also prime. See Bespoken for Sequences entry.
27386 is a product of 2 and a 4k+1 prime and thus it can be expressed as a sum of two squares in one way only viz. 119^2 + 115^2.
27385 is a product of two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 72^2 + 149^2 and 76^2 + 147^2.
27384 is a member of OEIS A187584: least number divisible by at least n of its digits, different and > 1, here n=5. The sequence can be generated as follows (permalink):
27383 is a member of OEIS A357262: numbers k such that the product of distinct digits of k equals the sum of the prime divisors of k, here 336. The sequence can be generated as follows (permalink):
27382 is a member of OEIS A064799: sum of n-th prime number and n-th composite number, here 3143 + 24239 are the 2697-th prime and non-prime numbers. See Bespoken for Sequences entry.
27381 is a member of OEIS A034818: concatenations p1, p2, p3 are all prime where (using | to represent concatenation) we have:
27380 is a product of a power of 2, a 4k+1 prime (5) and a 4k+1 prime (37) raised to the power of 2. Thus it can be expressed as a sum of two squares in three different ways viz. 22^2+164^2, 74^2+148^2 and 116^2+118^2.
27379 is a sphenic number whose sum of prime factors is a palindrome, here 161. See Bespoken for Sequences entry. The sequence members from 27379 up to 40000 are:
27378 is a product of 2, a 4k+3 prime (3) raised to an even power (4) and a 4k+1 prime (13) raised to the power 2 and so it can be expressed as a sum of two squares in exactly two ways viz. 63^2+153^2 =117^2+117^2.
27377 is a number that contains three 7s, is divisible by 7
27376 is a member of OEIS A107085: numbers n s.t. in decimal representation the largest digit is equal to the digital root, here 7. See entry for 27340.
27375 is a member of OEIS A059470: numbers that are the products of distinct substrings (>1) of themselves and do not end in 0, here 375 * 73 and 5 * 73 * 75. The sequence can be generated as follows (permalink):
27374 is a numbers whose arithmetic and multiplicative digital roots are not digits of the number itself but whose arithmetic root is equal to the number of digits in the number. Here the arithmetic digital root is 5 and the multiplicative digital root is 8. The number of digits in the number (5) corresponds to the arithmetic digital root.
27373 is a number n without the digit 0 with two distinct prime factors such that n + SOD(n) and n + POD(n) both have two distinct prime factors. Here SOD stands for sum of digits and POD for product of digits. Note that this is different to the arithmetic and multiplicative digital roots of a number. Here the results are 27395 = 5 * 5479 and 28255 = 5 * 5651 respectively. See record 27362.
27372 is a member of OEIS A070001: palindromic integers > 0, whose 'Reverse and Add!' trajectory (presumably) does not lead to another palindrome. See blog post 27372: Another Palindromic Day.
27371 is a member of OEIS A001607: a(n) = -a(n-1) - 2*a(n-2). The sequence can be generated as follows:
27370 is a member of OEIS A046387: products of 5 distinct primes. There are 237 such numbers in the range up to 40,000. The sequence can be generated as follows (permalink):
27369 is a product of a 4k+3 prime (3) raised to an even power (2) and a 4k+1 prime (3041). Thus it can be expressed as a sum of two squares in one way only viz. 165^2 + 12^2.
27368 is a composite number whose prime factors contain only the digits 1, 2 and 3. There are 849 such numbers in the range up to 40,000. See Bespoken for Sequences entry and blog post titled Quaternary Numbers.
27367 is a member of OEIS A124596: primes p such that q-p = 30, where q is the next prime after p, here 27397. This will occur on Saturday, April 6th 2024.
27366 is a member of OEIS A071927: barely abundant numbers: abundant n such that sigma(n)/n < sigma(m)/m for all m<n. See blog post Barely Abundant Numbers.
27365 is a sphenic number and product of primes that are the larger of twin prime pairs; sum of squares of number pairs. See blog post Triple Strength Sphenic Numbers And More. There are 166 such numbers in the range up to 40,000:
27364 is product of a power of 2 and a 4k+1 prime, thus it can be expressed as a sum of two squares in one way only viz. 160^2 + 42^2.
27363 is a number whose sum of digits about its centre point is the same, here 9. See Bespoken for Sequences entry. This number also has the property that the middle digit is equal to the arithmetic digital root, here 3. There are 270 five digit numbers with this property (in the range up to 40,000).
27362 is a product of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 131^2 + 101^2.
27361 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 156^2 + 55^2.
27360 is a member of OEIS A141586: strongly refactorable numbers n such that if n is divisible by d, it is divisible by the number of divisors of d. There are only 41 such numbers in the range up to 40,000. The sequence can be generated as follows:
27359 is a number with no repeating digits, whose additive digital and multiplicative digital roots are different from any of its digits and also from each other, here 8 and 0 respectively. There are 3386 such numbers in the range up to 40,000. See Bespoken for Sequences entry.
27358 is member of OEIS A094377: greatest number having exactly n representations as ab+ac+bc with 0 < a < b < c, here n=19 with a=5, b=134 and c=192 being one such representation (permalink). The initial terms are:
27357 is member of OEIS A187073: composite squarefree numbers whose average prime factor is a prime number, here 281. There are 1609 such numbers in the range up to 40,000. However, only 594 of these numbers are sphenic and of these, only 308 follow a 1, 2, 3 progression in terms of the number of digits in their prime factors. Of these 308, only 13 have prime factors with no digits in common (permalink). These are:
27356 is a member of OEIS A046411: composite numbers the concatenation of whose prime factors is a prime, here 227977. See blog post Thinning the Ranks.
27355 is a member of OEIS A046423: numbers requiring 3 steps to reach a prime under the prime factor concatenation procedure, here prime is 13195333. The sequence begins:
27354 is a number whose sum of digits about its centre point is the same, here 9. See Bespoken for Sequences entry. This number also has the property that the middle digit is equal to the arithmetic digital root of 5. There are 270 five digit numbers with this property (in the range up to 40,000). They can be found as follows (permalink):
27353 is a product of two 4k+1 primes and so it can be written as a sum of two squares in two different ways viz. 28^2+163^2 and 52^2+157^2.
27352 is a member of OEIS A172213: number of ways to place 4 non-attacking knights on a 4 X n board, here n=9. Initial members are:
27351 is a member of OEIS A181619: numbers k such that k^2+1 = 2*p,(k+1)^2+1 = 5*q, (k+2)^2+1 = 10*r where p, q, and r are primes. The sequence can be generated as follows (permalink):
27350 is a member of OEIS A023441: dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-11). The sequence can be generated as follows:
27349 is a member of OEIS A075282: interprimes which are of the form s*prime, s=7. There are 39 such interprimes in the range up to 40,000. The sequence can be generated as follows (permalink):
27348 is a number with no repeating digits whose arithmetic and multiplicative digital roots are equal but different to any of the digits of the number, here they equal 6. See Bespoken for Sequences entry.
27347 is a member of OEIS A323485: least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to k/n, here n=29. The sequence up to n=7 can be generated as follows (the numbers get ridiculously large for higher n but the algorithm works in principle):
27346 is a product of 2, a 4k+3 prime raised to an even power and a 4k+1 prime, thus it can be expressed as a sum of two squares in one way only viz. 165^2 + 11^2.
27345 is a member of OEIS A321022: the 100 terms of cycle that A321021 goes into using rule a(0)=0, a(1)=1 & a(n) = a(n-2)+a(n-1), keeping just digits that appear exactly once. The sequence can be generated as follows:
27344 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 140^2 + 88^2.
27343 is a member of OEIS A024670: numbers that are sums of two distinct positive cubes, here 30^3+7^3. The number is also a concatenation of two cubes, viz. 3^3 | 7^3. There are only a few numbers with this property in the range up to 40,000 and these are shown below. The sequence and related sequences can be generated as follows (permalink):
27342 is a member of OEIS A192087: potential magic constants of a 10 X 10 magic square composed of consecutive primes. See blog post Magic Constants Involving Prime Numbers. The initial members of the sequence are:
27341 is a semiprime whose average of prime factors is a perfect number, here (19+1439)/2 = 729 = 27^2. See Bespoken for Sequences entry. The semiprimes from 27341 to 40000 with this property are:
27340 is a member of OEIS A107085: numbers n such that in decimal representation the largest digit is equal to the digital root, here 7. The sequence can be generated as follows (permalink):
27339 is a sphenic number whose sum of prime factors is a palindrome, here 717. In the range up to 40000, there are 693 such numbers. See Bespoken for Sequences entry.
27338 is a product of 2 and a 4k+1 prime so it can be expressed as a sum of two squares in one way only viz. 143^2 + 83^2.
27337 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 164^2+21^2.
27336 is a member of OEIS A340639: number of regions inside a Reuleaux triangle formed by straight line segments mutually connecting all vertices and all points that divide sides into n equal parts, here n=10. The sequence begins:
27335 is a member of OEIS A020700: numbers k such that k + sum of its prime factors = (k+1) + sum of its prime factors. There are 37 such numbers in the range up to 40000. The sequence can be generated using the following code:
27334 is a sphenic number arising from OEIS A181622: sequence starting with 1 such that the sum of any two distinct terms has three distinct prime factors. This sequence begins:
27333 is a product of a 4k+3 prime (3) raised to an even power (2) and a 4k+1 prime (3037) and so it can be expressed as a sum of two squares in one way only viz. 162^2 + 33^2.
27332 is a product of a power of 2 and a 4k+1 prime, thus it can be expressed as a sum of two squares in one way only viz. 136^2 + 94^2.
27331 is a member of OEIS A048630: n-th 4k+1 prime times n-th 4k-1 prime, here n=19. The sequence can be generated as follows (permalink):
27330 is a member of OEIS A025004: a(1) = 3; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1. The sequence can be generated as follows (permalink):
27329 is a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 152^2 + 65^2.
27328 is a member of OEIS A004207: a(0) = 1, a(n) = sum of digits of all previous terms. The sequence can be generated as follows:
27327 is a number that is symmetric about its centre digit, here 27 - 3 - 27. There are 489 such numbers in the range up to 40000. See Bespoken for Sequences entry. Here are the members of the sequence from 27327 to 40000:
27326 is a member of OEIS A036301: numbers whose sum of even digits and sum of odd digits are equal. See blog post titled Odds and Evens: Statistics. How many "captives" are captured by the "attractor" 27326? The answer is 9 and the captives are 27299, 27307, 27309, 27311, 27320, 27321, 27322, 27324, 27328. Permalink.
27325 is a product of two 4k+1 prime factors, one raised to the power of 2, and so it can be expressed as the sum of two squares in three different ways viz. 10^2+165^2 and 91^2+138^2 and 107^2+126^2.
27324 is the lesser of a pair of adjacent composite numbers s.t. both are only one step away from their home primes (223331123 and 551093 respectively). See Bespoken for Sequences entry.
27323 is a member of OEIS A069107: composite numbers k that divide Fibonacci(k+1). The sequence can be generated as follows:
27322 is a sphenic number arising from the combination of two members of the sequence OEIS A181622, specifically 29 and 27293. See blog post Sphenic Generating Number Set.
27321 is a member of OEIS A130792: numbers n splittable into two parts which are seeds for a Fibonacci-like sequence containing n itself (here 273 and 21).
27320 is a number that is on the trajectory of 41 under the 17n+1 Collatz-like trajectory. The trajectory can be generated using the following algorithm:
27319 is a member of OEIS A064799: sum of n-th prime number and n-th composite number (3138 + 24181) with n=2692. See Bespoken for Sequences entry.
27318 is a member of OEIS A249951: numbers n such that 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8 is prime.
27317 is a member of OEIS A025005: a(1) = 5; a(n+1) = a(n)-th nonprime, where nonprimes begin at 1. The sequence can be generated as follows:
27316 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 154^2 + 60^2.
27315 is a member of OEIS A151745: composites that are the sum of two, three, four and five consecutive composite numbers. The numbers are:
27314 is a member of OEIS A075289: interprimes which are of the form s*prime, s=14. The sequence can be generated as follows:
27313 is a member of OEIS A363740: number of integer partitions of n containing a unique mode equal to the median, here n=22. The sequence can be generated from the following algorithm:
27312 is a member of OEIS A255215: numbers that belong to at least one amicable tuple, here (27312, 21168, 22200). The tuples are easy to find once one member is identified but otherwise, to find all the members of this sequence is very processor intensive.
27311 is a member of OEIS A178328: numbers k such that k^p-p is prime, where p is product of the digits of k. These primes are very large and the algorithm designed to generate the initial members of the sequence quickly stalls. Here are the initial members:
27310 is the sum of consecutive squares viz. 27^2 + 28^2 + ... + 46^2.
27309 is a member of OEIS A081848: number of numbers whose base-3/2 expansion (see A024629 where SageMath code be;pw can be found) has n digits, here n=25. The sequence can be generated up to 5394 after which the algorithm will time out on SageMathCell (permalink).
27308 is a member of OEIS A319142: total number of binary digits in the partitions of n into odd parts, here n=41. The sequence can be generated as follows:
27307 is a member of OEIS A048573: a(n) = a(n-1) + 2*a(n-2) with a(0)=2 and a(1)=3. The sequence can be generated as follows:
27306 is a product of 2, a 4k+3 prime raised to an even power and two 4k+1 primes and so it can be expressed as a sum of two squares in two different ways viz. 9^2+165^2 and 45^2+159^2.
27305 is a member of OEIS A084640: generalized Jacobsthal numbers given by a(n) = a(n-1) + 2*a(n-2) + 4 with a(0) = 0 and a(1) = 1. The sequence can be generated as follows:
27304 is a product of a power of 2 and a 4k+1 prime and so it can be expressed as a sum of two squares in one way only viz. 130^2 + 102^2.
27303 is a sphenic number whose three distinct prime factors have no digital root in common and whose digital root is different from the digital root of the original number. Note that 27305 has the same property. The sequence can be generated as follows (permalink):