A190899 Positive integers with recursively self-conjugate partitions.

1, 3, 4, 6, 9, 10, 11, 12, 15, 16, 17, 18, 21, 22, 24, 25, 27, 28, 31, 33, 34, 36, 37, 38, 40, 42, 43, 44, 45, 47, 48, 49, 51, 54, 55, 56, 57, 58, 59, 60, 61, 64, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106

Offset

1,2

Comments

A partition is self-conjugate if it is fixed under conjugation and it is recursively self-conjugate if it is self-conjugate and the portions below and to the right of its Durfee square are recursively self-conjugate. (See the Keith paper for a more detailed description.)

Only a finite number of positive integers do not have a recursively self-conjugate partition. The list is given in A190900.

Integers expressible as a_0^2 + 2*a_1^2 + ... + 2^k*a_k^2 with [a_0, a_1, .., a_k] a non-squashing partition. [See Keith link, p. 6]

Links

Table of n, a(n) for n=1..78.

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 not in the sequence.
The partition {4,4,2,2} of 12 is self-conjugate and is made up of Durfee squares thus 12 is 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 Durfree 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 not in the sequence.
Both self-conjugate partitions of 32 are not recursively so. Thus 32 is not 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 = 11}, TakeWhile[Union@ Flatten@ Array[Map[Total@ MapIndexed[#1^2*2^First[#2 - 1] &, #] &, f[#]] &, n], # <= n^2 &]] (* Michael De Vlieger, Oct 30 2018 *)

Crossrefs

Cf. A190900.

Sequence in context: A186709 A143037 A225059 * A173671 A325440 A080710

Adjacent sequences: A190896 A190897 A190898 * A190900 A190901 A190902

Keyword

nonn

Author

John W. Layman, May 23 2011

Status

approved