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 A330781 Numbers m that have recursively self-conjugate prime signatures. 1
 1, 2, 12, 36, 360, 27000, 75600, 378000, 1587600, 174636000, 1944810000, 5762988000, 42785820000, 5244319080000, 36710233560000, 1431699108840000, 65774855015100000, 731189187729000000, 1710146230392600000, 2677277333530800000, 2267653901500587600000, 115650348976529967600000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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) 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). EXAMPLE 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. 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}~Join~Take[#, FirstPosition[#, StringJoin["{", ToString[n], "}"]][] ][[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}}]]] ] &, Import["https://oeis.org/A330781/a330781.txt", "Data"] ] (* Michael De Vlieger, Jan 16 2020 *) CROSSREFS Cf. A006939, A025487, A181555, A181822, A181825, A322156. Sequence in context: A169630 A192385 A294464 * A185788 A305864 A324027 Adjacent sequences:  A330778 A330779 A330780 * A330782 A330783 A330784 KEYWORD nonn AUTHOR Michael De Vlieger, Jan 02 2020 STATUS approved

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Last modified August 11 06:25 EDT 2020. Contains 336422 sequences. (Running on oeis4.)