cp's OEIS Frontend

A239958 Number of partitions p of n such that (number of distinct parts of p) >= max(p) - min(p).

%I A239958 #13 Dec 31 2023 12:12:07
%S A239958 1,1,2,3,5,6,9,11,16,18,25,30,39,47,59,69,89,105,126,153,184,215,259,
%T A239958 307,362,426,501,583,687,800,923,1080,1252,1439,1666,1917,2202,2533,
%U A239958 2900,3311,3792,4326,4915,5605,6366,7205,8180,9259,10458,11815,13322
%N A239958 Number of partitions p of n such that (number of distinct parts of p) >= max(p) - min(p).
%F A239958 a(n) + A239954(n) = A000041(n) for n >= 0.
%e A239958 a(6) counts all of the 15 partitions of 7 except these 4:  61, 52, 511, 1111111.
%t A239958 z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
%t A239958 Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}]  (*A239954*)
%t A239958 Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
%t A239958 Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
%t A239958 Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}]  (*A034296*)
%t A239958 Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
%t A239958 ndpQ[p_]:=Module[{prt=Union[p]},Length[prt]>=(Max[prt]-Min[prt])]; Table[Length[Select[ IntegerPartitions[ n],ndpQ]],{n,0,50}] (* _Harvey P. Dale_, Dec 31 2023 *)
%Y A239958 Cf. A239954, A239955, A239956, A034296.
%K A239958 nonn,easy
%O A239958 0,3
%A A239958 _Clark Kimberling_, Mar 30 2014