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

1,6

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A085671 A126324 A035557 * A354151 A248947 A133916

Adjacent sequences: A305732 A305733 A305734 * A305736 A305737 A305738

Keyword

nonn

Author

Gus Wiseman, Jun 22 2018

Status

approved