OFFSET
1,1
COMMENTS
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.
LINKS
William J. Keith, Recursively Self-Conjugate Partitions, INTEGERS 11A, (2011) Article 12 (11 pages).
EXAMPLE
From Michael De Vlieger, Oct 23 2018: (Start)
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)
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,nice,fini,full
AUTHOR
John W. Layman, May 23 2011
STATUS
approved