Intended for: October 20, 2012
Timetable
- First draft entered by Alonso del Arte on October 19, 2011 ✓
- Draft reviewed by Daniel Forgues on October 19, 2012 ✓
- Draft to be approved by September 20, 2012
The line below marks the end of the <noinclude> ... </noinclude> section.
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 sum-of-divisors 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, ...}