Template:Sequence of the Day for October 20

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

Yesterday's SOTD * Tomorrow's SOTD

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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  ×  2 m ) σ  k 2 m

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 = (2 m +1 + 1) (2 m +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, ...}