A231919 a(n) = 3^n + (4^n - 3^n) * (d(n) - 3), where d(n) = A000005(n).

1, 2, -10, 81, -538, 4096, -12010, 65536, 19683, 1048576, -3840010, 49268766, -63920218, 268435456, 1073741824, 8546887871, -16921588858, 205383589230, -272553384010, 3291561314526, 4398046511104, 17592186044416, -70180457820010, 1406245165407356, 847288609443

Offset: 1

Author

Wesley Ivan Hurt, Nov 15 2013

Keywords

• sign
• easy
• less

Comments

a(n) is negative if and only if n is an odd prime (A065091). If n is prime, then a(n) = - A002250(n). If n is a semiprime (A001358), a(n) gives the n-th power of the number of divisors of n. For example, a(4) = d(4)^4 = 3^4 = 81. Similarly, a(6) = d(6)^6 = 4^6 = 4096.

Crossrefs

Cf. A000005, A002250.

Programs

Maple

with(numtheory); A231919:=n->3^n+(4^n-3^n)*(tau(n)-3); seq(A231919(n), n=1..100);

Mathematica

Table[3^n + (4^n - 3^n)(DivisorSigma[0,n] - 3), {n,100}]

Formula

a(n) = A000244(n) + A005061(n) * (A000005(n) - 3).