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A239958
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Number of partitions p of n such that (number of distinct parts of p) >= max(p) - min(p).
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4
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1, 1, 2, 3, 5, 6, 9, 11, 16, 18, 25, 30, 39, 47, 59, 69, 89, 105, 126, 153, 184, 215, 259, 307, 362, 426, 501, 583, 687, 800, 923, 1080, 1252, 1439, 1666, 1917, 2202, 2533, 2900, 3311, 3792, 4326, 4915, 5605, 6366, 7205, 8180, 9259, 10458, 11815, 13322
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OFFSET
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0,3
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LINKS
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FORMULA
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EXAMPLE
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a(6) counts all of the 15 partitions of 7 except these 4: 61, 52, 511, 1111111.
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MATHEMATICA
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z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}] (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}] (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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