A065405 Composite numbers k such that the sum of the divisors of k^2 is a prime.

4, 8, 27, 49, 64, 125, 169, 256, 289, 512, 529, 729, 841, 1849, 2197, 3125, 4913, 5329, 6241, 6889, 15625, 16129, 29791, 32768, 37249, 51529, 57121, 69169, 76729, 113569, 117649, 128881, 139129, 157609, 192721, 208849, 226981, 229441, 253009

Offset: 1

Author

Labos Elemer, Nov 06 2001

Keywords

• nonn

Comments

All these composite numbers k should be prime powers because if k=a*b with gcd(a,b)=1, then sigma(aabb) = sigma(aa)*sigma(bb) cannot be a prime; 46 of the 236 prime powers below 1000000 are here.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..70 from Harry J. Smith)

Crossrefs

Cf. A062700, A000203, A065403, A065404, A053182, A053183, A028982.

Programs

Mathematica

Select[ Range[3 10^5], ! PrimeQ[ # ] && PrimeQ[ DivisorSigma[1, #^2]] & ]

PARI

isok(k) = { !isprime(k) && isprime(sigma(k^2)) } \\ Harry J. Smith, Oct 18 2009

Formula

sigma(a(n)^2) = sigma(A065404(n)) = A065403(n) is prime.