A365615 a(n) = number of partitions p of n such that the least multiplicity of the parts of p is not a part of p.

1, 0, 2, 2, 2, 3, 5, 5, 9, 10, 15, 20, 25, 30, 41, 50, 61, 81, 99, 123, 154, 189, 231, 292, 346, 429, 526, 639, 759, 942, 1112, 1355, 1609, 1943, 2294, 2784, 3253, 3915, 4613, 5498, 6424, 7691, 8950, 10631, 12394, 14637, 17018, 20108, 23255, 27351, 31699

Offset

0,3

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A000041(n) - A365614(n).

Example

The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], having least multiplicities 1,1,1,1,1,1,5, respectively. The partitions that do not include least multiplicity as a part are [5], [3,2], and [1,1,1,1,1], so that a(5) = 3.

Mathematica

z = 40; f[n_] := f[n] = IntegerPartitions[n];
m[p_] := Min[Map[Length, Split[p]]]
Table[Count[f[n], p_ /; ! MemberQ[p, m[p]]], {n, 0, z}]

Crossrefs

Cf. A000041, A365613, A365614, A365616.

Sequence in context: A035658 A077018 A007918 * A278167 A239470 A320786

Adjacent sequences: A365612 A365613 A365614 * A365616 A365617 A365618

Keyword

nonn

Author

Clark Kimberling, Sep 17 2023

Status

approved