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.
The sums can be confirmed as follows (permalink):
number=27537
a=number.digits()
a=list(reversed(a))
def cyclic_perm(a):
n = len(a)
b = [[a[i - j] for i in range(n)] for j in range(n)]
return(b)
C=cyclic_perm(a)
total=0
for c in C:
subtotal=""
for i in range(len(c)):
subtotal+=str(c[i])
total+=int(subtotal)
print(total)
###################
number_s=37536
a=number_s.digits()
a=list(reversed(a))
def cyclic_perm(a):
n = len(a)
b = [[a[i - j] for i in range(n)] for j in range(n)]
return(b)
C=cyclic_perm(a)
total_s=0
for c in C:
subtotal=""
for i in range(len(c)):
subtotal+=str(c[i])
total_s+=int(subtotal)
print(total_s)
#####################
number_t=17592
a=number_t.digits()
a=list(reversed(a))
def cyclic_perm(a):
n = len(a)
b = [[a[i - j] for i in range(n)] for j in range(n)]
return(b)
C=cyclic_perm(a)
total_t=0
for c in C:
subtotal=""
for i in range(len(c)):
subtotal+=str(c[i])
total_t+=int(subtotal)
print(total_t)
OUTPUT
266664
266664
266664
The members of the sequence can be generated as follows (permalink):
INPUT
L=[]
for number in [1..100000]:
term=number
a=number.digits()
a=list(reversed(a))
def cyclic_perm(a):
n = len(a)
b = [[a[i - j] for i in range(n)] for j in range(n)]
return(b)
C=cyclic_perm(a)
sum1=0
for c in C:
s=""
for i in range(len(c)):
s+=str(c[i])
sum1+=int(s)
number=sigma(term)
a=number.digits()
a=list(reversed(a))
def cyclic_perm(a):
n = len(a)
b = [[a[i - j] for i in range(n)] for j in range(n)]
return(b)
C=cyclic_perm(a)
sum2=0
for c in C:
s=""
for i in range(len(c)):
s+=str(c[i])
sum2+=int(s)
number=euler_phi(term)
a=number.digits()
a=list(reversed(a))
def cyclic_perm(a):
n = len(a)
b = [[a[i - j] for i in range(n)] for j in range(n)]
return(b)
C=cyclic_perm(a)
sum3=0
for c in C:
s=""
for i in range(len(c)):
s+=str(c[i])
sum3+=int(s)
if sum1==sum2 and sum2==sum3:
L.append(term)
print(L)
OUTPUT
[1, 2907, 3339, 3726, 4293, 4371, 4614, 5049, 5319, 5607, 5751, 6291, 17901, 18009, 18441, 19413, 20349, 20655, 20943, 21219, 21267, 21573, 21627, 22137, 22191, 23355, 24831, 25647, 25731, 26019, 26145, 26163, 27405, 27537, 28035, 28215, 28227, 28305, 29601, 30429, 30447, 30609, 30645, 31059, 31074, 31392, 31455, 31671, 32049, 32103, 32265, 32913, 33318, 33354, 33507, 33831, 34155, 34443, 34515, 34611, 35055, 35235, 35343, 36018, 36045, 36231, 36261, 36423, 36603, 36801, 37233, 38502, 39093, 39402, 39903, 40146, 40254, 40365, 40482, 40851, 41094, 42117, 42603, 43029, 43227, 43575, 43605, 43713, 44091, 44109, 44361, 44658, 45297, 45315, 46143, 46734, 47274, 47277, 47463, 48057, 49131, 50271, 50445, 50463, 51381, 51543, 51705, 52533, 52809, 52863, 53433, 53457, 53667, 53919, 54027, 55809, 55941, 56151, 56481, 56619, 57129, 57507, 59607, 59751, 60129, 60741, 61287, 61533, 62577, 62883, 63153, 63771, 64881, 70467]
27537 is a sphenic and inconsummate number.
27537 can be rendered as a digit equation as follows:
2 + 7 = 5 -3 + 7
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