A241408 a(n) is the number of partitions of n such that the number of parts having multiplicity > 1 is a part.

0, 0, 1, 1, 2, 4, 5, 9, 11, 18, 24, 34, 46, 63, 83, 109, 147, 189, 245, 315, 406, 513, 650, 817, 1030, 1287, 1593, 1978, 2450, 3013, 3689, 4523, 5511, 6711, 8140, 9852, 11892, 14334, 17217, 20657, 24727, 29531, 35197, 41894, 49761, 59000, 69861, 82542, 97393

Offset: 0

Author

Clark Kimberling, Apr 22 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these 5 partitions: 411, 3111, 2211, 21111, 111111; e.g., the number of parts of 2211 that have multiplicity > 1 is 2, which counts 1 (with multiplicity 2) and 2 (also with multiplicity 2), so that 2211 is a term because 2 is a part.
		

Crossrefs

Cf. A241409, A241410, A241411, A241412.

Programs

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)