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