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A few preliminary pages (most of these need to be updated!)
Sequence of the Day for October 20
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.”
_______________
*
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, ...}