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

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

Offset: 1

Author

Michael De Vlieger, Jan 02 2020

Keywords

• nonn

Comments

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)

Examples

			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.
		

Links

• 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).

Crossrefs

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

Programs

Mathematica

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 *)