A326357 a(n) = number of partitions of n that occur more than once among the condensed partitions of n; see A239312.

0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 11, 11, 15, 19, 25, 29, 39, 47, 61, 67, 85, 107, 128, 154, 182, 219, 264, 310, 366, 441, 514, 608, 705, 836, 974, 1128, 1312, 1530, 1776, 2053, 2367, 2735, 3156, 3598, 4145, 4745, 5463, 6204, 7084, 8068, 9241, 10455, 11906, 13488

Offset: 0

Author

Clark Kimberling, Jun 30 2019

Keywords

• nonn
• easy

Examples

			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], with respective condensations [5], [4,1], [3,2], [3,2], [4,1], [3,2], [5]; of these, three occur more than once, so that a(5) = 3.
		

Crossrefs

Cf. A000009, A000041, A239312.

Programs

Mathematica

p[n_] := IntegerPartitions[n];
m[n_] := Map[Split, p[n]];
t[n_, k_] := Map[Total, m[n][[k]]];
u[n_] := Table[Sort[t[n, k], Greater], {k, 1, Length[p[n]]}]
v[n_] := Tally[u[n]];
w[n_] := Length[Select[v[n], #[[2]] > 1 &] ];l
Table[w[n], {n, 1, 20}]

Extensions

More terms from Alois P. Heinz, Jun 30 2019