This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A241408 #10 Jun 09 2022 21:49:23
%S A241408 0,0,1,1,2,4,5,9,11,18,24,34,46,63,83,109,147,189,245,315,406,513,650,
%T A241408 817,1030,1287,1593,1978,2450,3013,3689,4523,5511,6711,8140,9852,
%U A241408 11892,14334,17217,20657,24727,29531,35197,41894,49761,59000,69861,82542,97393
%N A241408 a(n) is the number of partitions of n such that the number of parts having multiplicity > 1 is a part.
%e A241408 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.
%t A241408 z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
%t A241408 Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}] (* A241408 *)
%t A241408 Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}] (* A241409 *)
%t A241408 Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
%t A241408 Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411 *)
%t A241408 Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412 *)
%Y A241408 Cf. A241409, A241410, A241411, A241412.
%K A241408 nonn,easy
%O A241408 0,5
%A A241408 _Clark Kimberling_, Apr 22 2014