A305735 Number of integer partitions of n whose greatest common divisor is a prime number.

0, 1, 1, 1, 1, 3, 1, 3, 2, 7, 1, 10, 1, 15, 8, 17, 1, 34, 1, 37, 16, 56, 1, 80, 6, 101, 27, 122, 1, 208, 1, 209, 57, 297, 20, 410, 1, 490, 102, 599, 1, 901, 1, 948, 194, 1255, 1, 1690, 14, 1985, 298, 2337, 1, 3327, 61, 3597, 491, 4565, 1, 6031, 1, 6842, 802

Offset: 1

Author

Gus Wiseman, Jun 22 2018

Keywords

• nonn

Examples

			The a(10) = 7 integer partitions are (82), (64), (622), (55), (442), (4222), (22222).
		

Links

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

Crossrefs

Cf. A000040, A000041, A000837, A018783, A100953, A143519, A289508, A302698, A305563, A305731.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],PrimeQ[GCD@@#]&]],{n,20}]

PARI

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

Formula

a(n) = Sum_{d|n} A143519(d) * A000041(n/d). - Andrew Howroyd, Jun 22 2018