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

1,2

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.

Links

Table of n, a(n) for n=1..47.

Example

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

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}]

Crossrefs

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

Sequence in context: A217254 A223488 A175686 * A368037 A373814 A351490

Adjacent sequences: A305560 A305561 A305562 * A305564 A305565 A305566

Keyword

nonn

Author

Gus Wiseman, Jun 05 2018

Status

approved