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A323034
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Where records occur in A321223.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}.
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MATHEMATICA
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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]]]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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