A322156 -id:A322156 - cp's OEIS frontend

A322457 Irregular triangle: Row n contains numbers k that have recursively symmetrical partitions having Durfee square with side length n.

1, 3, 4, 6, 10, 12, 9, 11, 15, 17, 21, 27, 16, 18, 22, 24, 28, 34, 36, 38, 40, 48, 25, 27, 31, 33, 37, 43, 45, 47, 49, 55, 57, 59, 61, 75, 36, 38, 42, 44, 48, 54, 56, 58, 60, 66, 68, 70, 72, 78, 80, 84, 86, 90, 108, 49, 51, 55, 57, 61, 67, 69, 71, 73, 79, 81

Offset: 1

Michael De Vlieger, Dec 11 2018

For all n, n^2 <= k <= 3*n^2.

For n > 5, some k may have more than 1 recursively self-conjugate partitions in the same row. For example, k = 90 in row 6 has two recursively self-conjugate partitions (RSCPs) with Durfee square of 6: (12,12,12,9,9,9,6,6,6,3,3,3) and (12,11,11,11,11,7,6,5,5,5,5,1). These RSCPs can be defined by dendritically laying out squares in the series {6,3,3} and {6,5,1} respectively.

			Triangle begins:
Row 1:   1,  3;
Row 2:   4,  6, 10, 12;
Row 3:   9, 11, 15, 17, 21, 27;
Row 4:  16, 18, 22, 24, 28, 34, 36, 38, 40, 48;
        ...
Row 2 contains the following recursively self-conjugate partitions with Durfee square with side length 2. Below are diagrams that place {2^0, 2^1, 2^2, ... 2^(m-1)} squares of side lengths in S = {k_1, k_2, k_3, ..., k_m}:
(2,2), sum 4, or in terms of squares, {2}:
   11
   11;
(3,2,1), sum 6, or in terms of squares, {2,1}:
   112
   11
   2;
(4,3,2,1), sum 10, or in terms of squares, {2,1,1}:
   1123
   113
   23
   3;
(4,4,2,2), sum 12, or in terms of squares, {2,2}:
   1122
   1122
   22
   22.

Michael De Vlieger, Table of n, a(n) for n = 1..10391 (rows 1 <= n <= 36, flattened)
Michael De Vlieger, Illustration for A322457
Michael De Vlieger, Annotated plot of k <= 1200 in rows n <= 34, vertical exaggeration 12x.

Cf.: A190899, A190900, A321223, A322156.

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]] ]; Array[Union@ Map[Total@ MapIndexed[#1^2*2^First[#2 - 1] &, #] &, f[#]] &, 7] // Flatten

First term of row n = n^2 = A000290(n).

Last term of row n = 3*n^2 = 3*A000290(n).

A323034 Where records occur in A321223.

Original entry on oeis.org

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

Michael De Vlieger, Jan 02 2019

nonn

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.

			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}.

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.

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

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]]]

A323035 Records in A321223.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 23, 25, 26, 28, 29, 30, 32, 34, 37, 41, 42, 48, 49, 50, 51, 56, 57, 59, 61, 68, 71, 72, 75, 76, 79, 80, 81, 82, 84, 86, 88, 89, 92, 93, 100, 103, 108, 118, 119, 120, 122, 125, 129, 130, 135, 141, 143

Offset: 1

Michael De Vlieger, Jan 04 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..381
William J. Keith, Recursively Self-Conjugate Partitions, Integers 11A, (2011) Article 12.

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

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 = 40, a}, 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]]; Union@ FoldList[Max, Array[Length[Lookup[a, #] /. k_ /; MissingQ@ k -> {}] &, Length@ a]]]

A330781 Numbers m that have recursively self-conjugate prime signatures.

Original entry on oeis.org

1, 2, 12, 36, 360, 27000, 75600, 378000, 1587600, 174636000, 1944810000, 5762988000, 42785820000, 5244319080000, 36710233560000, 1431699108840000, 65774855015100000, 731189187729000000, 1710146230392600000, 2677277333530800000, 2267653901500587600000, 115650348976529967600000

Offset: 1

Michael De Vlieger, Jan 02 2020

nonn

Let m be a product of a primorial, listed by A025487.
Consider the standard form prime power decomposition of m = Product(p^e), where prime p | m (listed from smallest to largest p), and e is the largest multiplicity of p such that p^e | m (which we shall hereinafter simply call "multiplicity").
Products of primorials have a list L of multiplicities in a strictly decreasing arrangement.
A recursively self-conjugate L has a conjugate L* = L. Further, elimination of the Durfee square and leg (conjugate with the arm) to leave the arm L_1. L_1 likewise has conjugate L_1* = L_1. We continue taking the arm, eliminating the new Durfee square and leg in this manner until the entire list L is processed and all arms are self-conjugate.
a(n) is a subsequence of A181825 (m with self-conjugate prime signatures).
Subsequences of a(n) include A006939 and A181555.
This sequence can be produced by a similar algorithm that pertains to recursively self-conjugate integer partitions at A322156.
From Michael De Vlieger, Jan 16 2020: (Start)
2 is the only prime in a(n).
The smallest 2 terms of a(n) are primorials, i.e., in A002110.
The smallest 5 terms of a(n) are highly composite, i.e., in A002182. (End)

			A025487(1) = 1, the empty product, is in the sequence since it is the product of no primes at all; this null sequence is self-conjugate.
A025487(2) = 2 = 2^1 -> {1} is self conjugate.
A025487(6) = 12 = 2^2 * 3 -> {2, 1} is self conjugate.
A025487(32) = 360 = 2^3 * 3^2 * 5 -> {3, 2, 1} is self-conjugate.
Graphing the multiplicities, we have:
3  x           3  x
2  x x   ==>   2  x x
1  x x x       1  x x x
   2 3 5          2 3 5
where the vertical axis represents multiplicity and the horizontal the k-th prime p, and the arrow represents the transposition of the x's in the graph. We see that the transposition does not change the prime signature (thus, m is also in A181825), and additionally, the prime signature is recursively self-conjugate.

Michael De Vlieger, Table of n, a(n) for n = 1..2243
Michael De Vlieger, A322156 encoding for a(n) for n = 1..75047, with the largest term a(75047) = A002110(60)^60 (8019 decimal digits).
Michael De Vlieger, Indices of terms in a(n) that are also in the Chernoff Sequence (A006939)
Michael De Vlieger, Indices of terms m in a(n) that are also in A181555 = A002110(n)^n (useful for assuring a(n) <= m is complete).

Cf. A006939, A025487, A181555, A181822, A181825, A322156.

Block[{n = 6, f, g}, 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]] ]; g[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}}]]] ]; {1}~Join~Take[#, FirstPosition[#, StringJoin["{", ToString[n], "}"]][[1]] ][[All, 1]] &@ Sort[MapIndexed[{Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #2], ToString@ #1} & @@ {#1, g[#1], First@ #2} &, Apply[Join, Array[f[#] &, n] ] ] ] ]
(* Second program: decompress dataset of a(n) for n = 0..75047 *)
{1}~Join~Map[Block[{k, w = ToExpression@ StringSplit[#, " "]}, k = Total@ w; Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, 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}}]]] ] &, Import["https://oeis.org/A330781/a330781.txt", "Data"] ] (* Michael De Vlieger, Jan 16 2020 *)

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A322457 Irregular triangle: Row n contains numbers k that have recursively symmetrical partitions having Durfee square with side length n.

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A323034 Where records occur in A321223.

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A323035 Records in A321223.

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A330781 Numbers m that have recursively self-conjugate prime signatures.

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