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.

27684 is a number that maps back to itself after two or more repetitions of the prime / non-prime algorithm. See blog post Prime and Non-Prime Digit Sequence. Here the trajectory is 27684 --> 27675 --> 27690 --> 27684 and thus 27684 is a vortical and thus part of the vortex beginning and ending with 27684.

27684 has no repeating digits and its arithmetic digital root (2) and its multiplicative digital root (0) are different from any of the digits of the number. The next number, 27685, also has this property. See Bespoken for Sequences entry.

27684 is a tau number because it is divisible by the number of its divisors (18).

27684 is a hoax number since the sum of its digits (27) coincides with the sum of the digits of its distinct prime factors.

27684 is an untouchable, abundant, pseudoperfect but NOT a Zumkeller number.

27684 can be rendered as a digit equation as follows:

(2 ^ (7 - 6)) | 8 = 4