A190900 Positive integers without recursively self-conjugate partitions.

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

Offset: 1

Author

John W. Layman, May 23 2011

Keywords

• nonn
• nice
• fini
• full

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.

Examples

			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)
		

Links

• William J. Keith, Recursively Self-Conjugate Partitions, INTEGERS 11A, (2011) Article 12 (11 pages).

Crossrefs

Cf. A190899.

Programs

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 *)