A181822 -id:A181822 - cp's OEIS frontend

A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

1, 2, 12, 36, 120, 360, 1680, 5040, 5400, 27000, 36960, 75600, 110880, 378000, 960960, 1587600, 1663200, 2882880, 7938000, 8316000, 32672640, 34927200, 43243200, 98017920, 174636000, 216216000, 277830000, 908107200, 1152597600, 1241560320, 1470268800, 1944810000

Offset: 1

Matthew Vandermast, Dec 08 2010

A025487(n) is included iff A025487(n) = A181822(n).

Closed under the binary operations of GCD and LCM, since a self-conjugate partition of Omega(a(n)) (which the prime signature of these numbers is) is the concatenation of self-conjugate hooks of decreasing size while moving downward and to the right in the Ferrers diagram, and the GCD (or LCM) of two terms a(i) and a(j) is obtained by taking the smaller (or larger, respectively) of the corresponding hooks. For example, GCD(a(8),a(11)) = GCD(5040,36960) = 1680 = a(7), and LCM(a(8),a(11)) = 110880 = a(13). The two binary operations make the set {a(n)} into a lattice order. - Richard Peterson, May 29 2020

			A025487(11) = 36 = 2^2*3^2 has a prime signature of (2,2), which is a self-conjugate partition; hence, 36 is included in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11879 (first 578 terms from Amiram Eldar, terms <= 10^70)
David A. Corneth, PARI-program
Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Cf. A025487, A181822, A303557.
Includes subsequences A006939 and A181555.
See also A181823, A181824, A181826, A181827.

\\ See Corneth link \\ David A. Corneth, Jun 03 2020

a(18)-a(32) from Amiram Eldar, Jan 19 2019

A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 7, 6, 5, 7, 4, 6, 6, 8, 7, 6, 8, 5, 7, 7, 9, 8, 7, 9, 6, 8, 6, 8, 10, 7, 9, 6, 8, 8, 10, 7, 9, 7, 9, 11, 8, 10, 5, 7, 9, 9, 11, 8, 10, 8, 10, 12, 9, 11, 6, 8, 10, 8, 10, 12, 7, 9, 9, 11, 9, 8, 11, 10, 13, 10, 12, 7, 9, 11, 9, 11, 13, 8, 10, 10, 12

Offset: 1

Alford Arnold

			a(3) = 2 since A025487(3) = 4 = 2*2; a(5) = 3 since A025487(5) = 8 = 2*2*2; ...

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A025487, A000041, A035796, A036042, A001222.

a(n) = A001222(A025487(n)) = A001222(A181822(n)).

More terms from Henry Bottomley, Apr 30 2001
Edited to accommodate change in A025487's offset by Matthew Vandermast, Nov 08 2008
Definition corrected by Álvar Ibeas, Nov 01 2014

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 32, 12, 64, 24, 36, 128, 48, 72, 256, 96, 144, 30, 512, 192, 216, 288, 60, 1024, 384, 432, 576, 120, 2048, 768, 864, 180, 1152, 240, 1296, 4096, 1536, 1728, 360, 2304, 480, 2592, 8192, 3072, 3456, 720, 900, 4608, 960, 5184, 1080, 16384

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

The number of second signatures represented by the divisors of A181800(n) equals the number of prime signatures represented among the divisors of a(n). Cf. A212172, A212644.

A permutation of A025487.

			6 (whose prime factorization is 2*3) is the largest squarefree divisor of 144 (whose prime factorization is 2^4*3^2). Since 144 = A181800(10), and 144/6 = 24, a(10) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A003557, A007947, A085082, A181800, A212172, A212642, A212644.

Other permutations of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.

a(n) = A003557(A181800(n)).

A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 7, 4, 6, 6, 9, 7, 7, 9, 11, 10, 8, 12, 9, 13, 5, 10, 13, 9, 15, 14, 15, 9, 14, 16, 10, 18, 19, 17, 13, 18, 10, 19, 11, 16, 21, 12, 15, 24, 19, 17, 22, 16, 22, 12, 23, 24, 6, 19, 20, 29, 21, 21, 26, 22, 25, 13, 30, 27, 11, 26, 25, 19, 34

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Also the number of members of A025487 that divide A025487(n).

			5 members of A025487 divide A025487(6) = 12 (namely, 1, 2, 4, 6 and 12); therefore, a(6) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A085082, A025487, A181822, A322584.
Rearrangement of A115728, A115729 and A238690.
A116473(n) is the number of times n appears in the sequence.

lpsQ[n_] := n == 1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1, 1]] == Prime[Length[f]]); lps = Select[Range[6000], lpsQ]; c[n_] := Count[Divisors[n], ?(MemberQ[lps, #] &)]; c /@ lps  (* _Amiram Eldar, Jan 21 2024 *)

a(n) = A085082(A025487(n)) = A085082(A181822(n)).

a(n) = A322584(A025487(n)). - Amiram Eldar, Jan 21 2024

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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A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

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A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

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A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

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A238746 Number of distinct prime signatures that occur among the divisors of the n-th prime signature number (A025487(n)).

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

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