A023194: Numbers such that is prime.

{ 2, 4, 9, 16, 25, 64, 289, 729, 1681, ... }
In 2005, Zak Seidov wondered why all terms except the first are squares.* Gabe Cunningham provided the answer:
“From the fact that (...) the sumofdivisors function is multiplicative, we can derive that is even except when is a square or twice a square.”
“If , that is, is twice an odd square, then σ (n) = 3 σ ((2 k + 1) 2 )  . If , that is, is twice an even square, then is only prime if is a power of 2; otherwise we have for some positive integer .”
“So the only possible candidates for values of other than squares such that is prime are odd powers of 2. But σ (2 2 m +1) = 2 2 m +2 − 1 = (2 m +1 + 1) (2 m +1 − 1)  , which is only prime when , that is, when . So 2 is the only nonsquare such that is prime.”
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* A055638 Numbers for which is prime: {2, 3, 4, 5, 8, 17, 27, 41, 49, 59, 64, 71, 89, 101, 125, 131, 167, 169, 173, 256, 289, ...}
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Sequences in the News
 Feb 01 2018 Alphabet announced a $8,589,869,056 = $A000396(6) stock buyback.
 Jan 03 2018 Largest known term of A000043 announced: 77232917.
 Nov 18 2016 PrimeGrid proves that 10223 is not a Sierpinski number, since 10223 × 2 31172165 + 1 is prime. So no changes to A076336 for now.
 Sep 14 2016 Tom Greer discovers the twin primes 2996863034895 × 2 1290000 ± 1 using PrimeGrid, TwinGen and LLR.
 Jan 19 2016 Largest known term of A000043 announced: 74207281, also discovered by Curtis Cooper.
 Mar 02 2014 Fredrik Johansson announces a computation of the partition number p(10 20) ≈ 1.8381765 × 10 11140086259, the largest known term of A000041.
 Dec 06 2013 Microsoft launches a challenge to find large nonMersenne primes, A138837.
