A305563 Number of reducible integer partitions of n.

1, 2, 3, 4, 7, 7, 15, 16, 27, 30, 56, 56, 100, 105, 157, 188, 287, 303, 470, 524, 724, 850, 1197, 1339, 1856, 2135, 2814, 3305, 4360, 4951, 6532, 7561, 9563, 11195, 14165, 16328, 20631, 23866, 29471, 34320, 42336, 48672, 59872, 69139, 83625, 96911, 117153

Offset: 1

Author

Gus Wiseman, Jun 05 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 reducible if either m is of size 1 or gcd(m_1, ..., m_k) = 1 and the multiset {y_1, ..., y_k} is also reducible.

Examples

			The a(6) = 7 reducible integer partitions are (6), (51), (411), (321), (3111), (21111), (111111). Missing from this list are (42), (33), (222), (2211).
		

Crossrefs

Cf. A007916, A071625, A181819, A182850, A182857, A275870, A304465, A304660, A304687, A304818, A305564, A305565, A305566.

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}]