A305736 Number of integer partitions of n whose greatest common divisor is composite (nonprime and > 1).

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 4, 0, 1, 1, 5, 0, 4, 0, 8, 1, 1, 0, 14, 1, 1, 3, 16, 0, 10, 0, 22, 1, 1, 1, 41, 0, 1, 1, 45, 0, 18, 0, 57, 9, 1, 0, 94, 1, 8, 1, 102, 0, 38, 1, 138, 1, 1, 0, 221, 0, 1, 17, 231, 1, 59, 0, 298, 1, 22

Offset: 1

Author

Gus Wiseman, Jun 22 2018

Keywords

• nonn

Examples

			The a(12) = 4 integer partitions are (12), (8 4), (6 6), (4 4 4).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000837, A018783, A100953, A289508, A302698, A305563, A305731, A305735.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!(GCD@@#==1||PrimeQ[GCD@@#])&]],{n,0,20}]

PARI

seq(n)={dirmul(vector(n, n, numbpart(n)), dirmul(vector(n, n, moebius(n)), vector(n, n, n>1&&!isprime(n))))} \\ Andrew Howroyd, Jun 22 2018

Formula

a(n) = A018783(n) - A305735(n). - Andrew Howroyd, Jun 22 2018