A065814 a(n) = tau(n)^2 - tau(n^2), where tau(n) = A000005(n).

0, 1, 1, 4, 1, 7, 1, 9, 4, 7, 1, 21, 1, 7, 7, 16, 1, 21, 1, 21, 7, 7, 1, 43, 4, 7, 9, 21, 1, 37, 1, 25, 7, 7, 7, 56, 1, 7, 7, 43, 1, 37, 1, 21, 21, 7, 1, 73, 4, 21, 7, 21, 1, 43, 7, 43, 7, 7, 1, 99, 1, 7, 21, 36, 7, 37, 1, 21, 7, 37, 1, 109, 1, 7, 21, 21, 7, 37, 1, 73, 16, 7, 1, 99, 7, 7, 7, 43

Offset: 1

Author

Labos Elemer, Nov 22 2001

Keywords

• nonn

Comments

If n = p^c = power of a prime, then a(n) = (c+1)^2 - (2c+1) = c^2. If n is squarefree with k prime factors then a(n) = 4^k - 3^k, so A065814(A002110(n)) = 4^n - 3^n = A005061(n). Terms depend on prime signature only.

If n is a prime (A000040), then a(n) = 1. If n is a semiprime (A001358), then a(n) = 4 + 3*ceiling(sqrt(n)) - 3*floor(sqrt(n)). If n is a triprime (A014612), then a(n) = 9 * floor(1/omega(n)) + 21 * (1 - (omega(n) mod 2)) + 37 * floor(omega(n)/3), n > 1. - Wesley Ivan Hurt, May 24 2013

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A000040, A000290, A000961, A001358, A002110, A005061, A014612, A035116, A048691.

Programs

Mathematica

a[n_] := DivisorSigma[0, n]^2 - DivisorSigma[0, n^2]; Array[a, 100] (* Amiram Eldar, Apr 25 2024 *)

PARI

a(n) = { numdiv(n)^2 - numdiv(n^2) } \\ Harry J. Smith, Oct 31 2009

Formula

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

G.f.: Sum_{n>=1} A000005(n^2)*x^(2*n)/(1-x^n). - Mircea Merca, Feb 26 2014

a(n) = A035116(n) - A048691(n). - Amiram Eldar, Apr 25 2024