A323034 Where records occur in A321223.

1, 27, 103, 175, 198, 310, 411, 495, 627, 675, 720, 838, 880, 1008, 1014, 1191, 1245, 1296, 1575, 1776, 1911, 1953, 2011, 2136, 2160, 2416, 2502, 2673, 2736, 3015, 3123, 3195, 3270, 3450, 3528, 3600, 3696, 4041, 4248, 4251, 4323, 4356, 4410, 4518, 4531, 4716

Offset: 1

Author

Michael De Vlieger, Jan 02 2019

Keywords

• nonn

Comments

Numbers k that set records for the number m of recursively self-conjugate partitions (RCSPs).

1 is the only square in the sequence.

The graph of A321223 suggests there is a finite number of numbers k with a given number m of RSCPs (not all such k appear here). We know that A190900 (positive integers without RSCPs) is finite. For index i <= 2^16, there are 6 squares in A321223, i.e., those of {1, 2, 3, 5, 8}, that have just 1 RSCP; there are 120 nonsquares 3 <= k <= 590 in A321223 that have m = 1 RSCP. In the same range, there are 127 numbers 27 <= k <= 830 in A321223 that have m = 2 RSCPs, and 142 numbers 103 <= k <= 1280 in A321223 that have m = 3 RSCPs. This sequence includes many of the first terms k of these finite sequences, all k having m RSCPs.

Examining the smallest 381 terms (i.e., all k < 2^16) and the plot of A321223, we observe the following:

1. a(3) = 103 and a(23) = 2011 are the only primes.

2. a(2) = 27 = 3^3 and a(64) = 6561 = 3^8 are the only prime powers.

3. Numbers k such that k mod 3 = 2 are never in this sequence.

4. Only k in {1, 103, 175, 310, 838, 880, 2011, 2416, 4531, 4720, 5872, 11248, 11632, 12400, 15136, 16081, 19696, 20464, 29296, 40816, 51568, 52336} are congruent to 1 (mod 3); this of course includes both primes 103 and 2011. It appears that there are yet more k congruent to 1 (mod 3) greater than 2^16.

Examples

			RSCPs of the first 3 terms:
  a(1) = 1:   (1).
  a(2) = 27:  (6,6,6,3,3,3), (6,5,5,5,5,1).
  a(3) = 103: (13,13,13,10,10,10,7,6,6,6,3,3,3),
              (13,12,12,12,12,8,7,6,5,5,5,5,1),
              (13,12,12,10,9,9,9,9,9,4,3,3,1).
RSCPs stated in terms of recursive Durfee squares for the first 5 terms:
  a(1) = 1:   {1}.
  a(2) = 27:  {3,3}, {5,1}.
  a(3) = 103: {7,3,3}, {7,5,1}, {9,3,1}.
  a(4) = 175: {9,5,3,1}, {11,3,3}, {11,5,1}, {13,1,1}.
  a(5) = 198: {10,5,2,2}, {10,7}, {12,3,3}, {12,5,1}, {14,1}.
  a(6) = 310: {12,7,3,2}, {12,9,1}, {14,5,4}, {14,7,2},
              {16,3,3}, {16,5,1}.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..381
• William J. Keith, Recursively Self-Conjugate Partitions, Integers 11A, (2011) Article 12.
• Michael De Vlieger, Illustration of first 5 terms.
• Michael De Vlieger, Recursively Self-Conjugate Partitions for Numbers in OEIS A323034, 3600 x 2400 pixel PNG file.
• Michael De Vlieger, Plot of terms of A323034 (black) in A321223(n) (color) for n <= 65536.

Crossrefs

Cf. A190899, A190900, A321223, A322156, A322457.

Programs

Mathematica

f[w_] := Block[{k}, k = Total@ w; Total@ Map[Apply[Function[{s, t}, s Array[Boole[t <= # <= s + t - 1] &, k] ], #] &, Apply[Join, Prepend[Table[Function[{v, c}, Map[{w[[k]], # + 1} &, Map[Total[v #] &, Tuples[{0, 1}, {Length@ v}]]]] @@ {Most@ #, ConstantArray[1, Length@ # - 1]} &@ Take[w, k], {k, 2, Length@ w}], {{w[[1]], 1}}]]] ]; g[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]] ]; Block[{n = 30, a, s}, a = Merge[Map[<| #1 -> #2 |> & @@ # &, #], Identity] &@ TakeWhile[Sort@ Map[{Total@ #2, #1, #2} & @@ {#, f[#]} &, Apply[Join, Array[g, n]] ], First@ # <= n^2 &][[All, 1 ;; 2]]; s = Array[Length[Lookup[a, #] /. k_ /; MissingQ@ k -> {}] &, Length@ a]; Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]