A065773 Number of divisors of square of true prime powers arising in A065405.

5, 7, 7, 5, 13, 7, 5, 17, 5, 19, 5, 13, 5, 5, 7, 11, 7, 5, 5, 5, 13, 5, 7, 31, 5, 5, 5, 5, 5, 5, 13, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 7, 7, 5, 5, 5, 5, 5, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..500

Formula

a(n) = A000005(A065405(n)^2).

If A065405(n) = q^c, a prime-power, then sigma(q^(2c)) = A000203(q^(2c)) = (-1 + q^(2c+1))/(q-1) = (-1 + q^A000005(A065405(n)^2))/(q-1) also a prime, from A065403.

Example

For k = 3125, tau(k^2) = 11, sigma(k^2) = 12207031 = (5^(tau(k^2)) - 1)/4 = A065403(16) is also a prime.

Mathematica

DivisorSigma[0, Select[Range[10^5], ! PrimeQ[#] && PrimeQ[DivisorSigma[1, #^2]] &]^2] (* Amiram Eldar, Jan 31 2025 *)

Prog

(PARI) { n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), a=numdiv(m^2); write("b065773.txt", n++, " ", a); if (n==100, return)) ) } \\ Harry J. Smith, Oct 30 2009

Crossrefs

Cf. A000005, A065405, A000203, A065403, A065771, A065772, A025475.

Sequence in context: A393106 A011269 A093723 * A114916 A001989 A086056

Adjacent sequences: A065770 A065771 A065772 * A065774 A065775 A065776

Keyword

nonn

Author

Labos Elemer and Robert G. Wilson v, Nov 19 2001

Status

approved