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A190899 Positive integers with recursively self-conjugate partitions. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 7 14:09 EDT 2021. Contains 343650 sequences. (Running on oeis4.)