This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A190900 #23 Oct 31 2018 04:38:48
%S A190900 2,5,7,8,13,14,19,20,23,26,29,30,32,35,39,41,46,50,52,53,62,63,65,74,
%T A190900 77,92,95,104,107,109,110,116,119,128,158,159,170,173,182,185,221,248,
%U A190900 251,317,545
%N A190900 Positive integers without recursively self-conjugate partitions.
%C A190900 Numbers with recursively self-conjugate partitions are given in A190899. See that sequence or the Keith reference for more details.
%C A190900 It is proved in the reference that this list is exhaustive.
%H A190900 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 A190900 From _Michael De Vlieger_, Oct 23 2018: (Start)
%e A190900 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.
%e A190900 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.
%e A190900 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.
%e A190900 Both self-conjugate partitions of 32 are not recursively so. Thus 32 is in the sequence. (End)
%t A190900 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 *)
%Y A190900 Cf. A190899.
%K A190900 nonn,nice,fini,full
%O A190900 1,1
%A A190900 _John W. Layman_, May 23 2011