

A190899


Positive integers with recursively selfconjugate 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
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OFFSET

1,2


COMMENTS

A partition is selfconjugate if it is fixed under conjugation and it is recursively selfconjugate if it is selfconjugate and the portions below and to the right of its Durfee square are recursively selfconjugate. (See the Keith paper for a more detailed description.)
Only a finite number of positive integers do not have a recursively selfconjugate 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 nonsquashing partition. [See Keith link, p. 6]


LINKS

Table of n, a(n) for n=1..78.
William J. Keith, Recursively SelfConjugate 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 selfconjugate, thus 5 is not in the sequence.
The partition {4,4,2,2} of 12 is selfconjugate 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 selfconjugate. We eliminate the Durfee square {4,4,4,4} which leaves us with {4,1,1,1} which is selfconjugate, but when we eliminate the Durfree square {1} from this, we are left with {1,1,1} which is not selfconjugate. There are no other selfconjugate partitions of 30, therefore 30 is not in the sequence.
Both selfconjugate 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



