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A190900
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Positive integers without recursively self-conjugate partitions.
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6
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2, 5, 7, 8, 13, 14, 19, 20, 23, 26, 29, 30, 32, 35, 39, 41, 46, 50, 52, 53, 62, 63, 65, 74, 77, 92, 95, 104, 107, 109, 110, 116, 119, 128, 158, 159, 170, 173, 182, 185, 221, 248, 251, 317, 545
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OFFSET
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1,1
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COMMENTS
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Numbers with recursively self-conjugate partitions are given in A190899. See that sequence or the Keith reference for more details.
It is proved in the reference that this list is exhaustive.
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LINKS
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EXAMPLE
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None of the partitions of 5, {{5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {1,1,1,1,1}} are self-conjugate, thus 5 is in the sequence.
The partition {4,4,2,2} of 12 is self-conjugate and is made up of Durfee squares thus 12 is not in the sequence.
The partition {8,5,5,5,4,1,1,1} of 30 is self-conjugate. We eliminate the Durfee square {4,4,4,4} which leaves us with {4,1,1,1} which is self-conjugate, but when we eliminate the Durfee square {1} from this, we are left with {1,1,1} which is not self-conjugate. There are no other self-conjugate partitions of 30, therefore 30 is in the sequence.
Both self-conjugate partitions of 32 are not recursively so. Thus 32 is in the sequence. (End)
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MATHEMATICA
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f[n_] := Block[{w = {n}, c}, c[x_] := Apply[Times, Most@ x - Reverse@ Accumulate@ Reverse@ Rest@ x]; Reap[Do[Which[And[Length@ w == 2, SameQ @@ w], Sow[w]; Break[], Length@ w == 1, Sow[w]; AppendTo[w, 1], c[w] > 0, Sow[w]; AppendTo[w, 1], True, Sow[w]; w = MapAt[1 + # &, Drop[w, -1], -1] ], {i, Infinity}] ][[-1, 1]] ]; With[{n = 30}, Complement[Range@ Last@ #, #] &@ TakeWhile[Union@ Flatten@ Array[Map[Total@ MapIndexed[#1^2*2^First[#2 - 1] &, #] &, f[#]] &, n], # <= n^2 &]] (* Michael De Vlieger, Oct 30 2018 *)
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CROSSREFS
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KEYWORD
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nonn,nice,fini,full
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AUTHOR
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STATUS
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approved
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