1,1

All numbers except the first are squares. Why? - Zak Seidov (zakseidov(AT)yahoo.com), Jun 10 2005

Answer from Gabe Cunningham (gcasey(AT)MIT.EDU): "From the fact that the sigma (the sum-of-divisors function) is multiplicative, we can derive that the sigma(n) is even except when n is a square or twice a square.

"If n = 2(2k+1)^2, that is, n is twice an odd square, then sigma(n) = 3*sigma((2k+1)^2). If n = 2(2k)^2, that is, n is twice an even square, then sigma(n) is only prime if n is a power of 2; otherwise we have sigma(n) = sigma(8*2^m) * sigma(k/2^m) for some positive integer m.

"So the only possible candidates for values of n other than squares such that sigma(n) is prime are odd powers of 2. But sigma(2^(2m+1)) = 2^(2m+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 non-square n such that sigma(n) is prime."

David W. Wilson, Table of n, a(n) for n = 1..10000

Select[ Range[ 100000 ], PrimeQ[ DivisorSigma[ 1, # ] ]& ]

(PARI) for(x=1, 1000, if(isprime(sigma(x)), print(x))) (Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 23 2004)

Cf. A055638.

Cf. A107926.

Sequence in context: A006474 A110878 A077137 this_sequence A114080 A090676 A000291

Adjacent sequences: A023191 A023192 A023193 this_sequence A023195 A023196 A023197

nonn,easy,nice

David W. Wilson (davidwwilson(AT)comcast.net)