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A few preliminary pages (most of these need to be updated!)

Sequence of the Day for October 20

A023194: Numbers
n
such that
σ (n)
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 sum-of-divisors function is multiplicative, we can derive that
σ (n)
is even except when
n
is a square or twice a square.”
“If
n = 2 (2 k + 1) 2
, that is,
n
is twice an odd square, then
σ (n) = 3 σ ((2 k + 1) 2 )
. If
n = 2 (2 k) 2
, that is,
n
is twice an even square, then
σ (n)
is only prime if
n
is a power of 2; otherwise we have
σ (n) = σ (8  ×  2m ) σ
k
2m
for some positive integer
m
.”
“So the only possible candidates for values of
n
other than squares such that
σ (n)
is prime are odd powers of 2. But
σ (2 2 m +1) = 2 2 m +2  −  1 = (2m +1 + 1) (2m +1  −  1)
, which is only prime when
m = 0
, that is, when
n = 2
. So 2 is the only nonsquare
n
such that
σ (n)
is prime.”

_______________

* A055638 Numbers
n
for which
σ (n 2 )
is prime:
{2, 3, 4, 5, 8, 17, 27, 41, 49, 59, 64, 71, 89, 101, 125, 131, 167, 169, 173, 256, 289, ...}