A305731 Number of irreducible integer partitions of n.

1, 0, 0, 0, 1, 0, 4, 0, 6, 3, 12, 0, 21, 1, 30, 19, 43, 10, 82, 20, 103, 68, 152, 58, 236, 102, 301, 196, 413, 205, 653, 310, 788, 580, 1115, 718, 1649, 1006, 2149, 1714, 3018, 2247, 4502, 3389, 6036, 5509, 8647, 7601, 12678, 11310, 17541

Offset: 0

Author

Gus Wiseman, Jun 22 2018

Keywords

• nonn

Comments

A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is irreducible if m is of size > 1 and either gcd(m_1, ..., m_k) > 1 or the multiset {y_1, ..., y_k} is irreducible.

Examples

			The a(6) = 4 irreducible partitions are (42), (33), (222), (2211).
		

Crossrefs

Cf. A100953, A181819, A182850, A182857, A275870, A304818, A305563, A305733, A305735.

Programs

Mathematica

ptnredQ[y_]:=Or[Length[y]==1,And[GCD@@y==1,ptnredQ[Sort[Length/@Split[y],Greater]]]];
Table[Length[Select[IntegerPartitions[n],!ptnredQ[#]&]],{n,20}]