A190899 - OEIS

%I #27 Nov 04 2018 00:28:07

%S 1,3,4,6,9,10,11,12,15,16,17,18,21,22,24,25,27,28,31,33,34,36,37,38,

%T 40,42,43,44,45,47,48,49,51,54,55,56,57,58,59,60,61,64,66,67,68,69,70,

%U 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

%N Positive integers with recursively self-conjugate partitions.

%C 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.)

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

%C 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]

%H William J. Keith, <a href="http://math.colgate.edu/~integers/vol11a.html">Recursively Self-Conjugate Partitions</a>, INTEGERS 11A, (2011) Article 12 (11 pages).

%e From _Michael De Vlieger_, Oct 23 2018: (Start)

%e 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.

%e The partition {4,4,2,2} of 12 is self-conjugate and is made up of Durfee squares thus 12 is in the sequence.

%e 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.

%e Both self-conjugate partitions of 32 are not recursively so. Thus 32 is not in the sequence. (End)

%t 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 *)

%Y Cf. A190900.

%K nonn

%O 1,2

%A _John W. Layman_, May 23 2011