A181555 -id:A181555 - cp's OEIS frontend

A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

1, 2, 4, 12, 24, 36, 48, 120, 240, 360, 720, 1680, 5040, 10080, 15120, 20160, 25200, 45360, 50400, 110880, 221760, 332640, 554400, 665280, 2882880, 8648640, 14414400, 17297280, 43243200, 294053760

Offset: 1

Matthew Vandermast, May 22 2012

Sequence can be used to find the largest highly composite number in subsequences of A212165 (of which there are several in the database).
Ramanujan showed that, in the canonical prime factorization of a highly composite number with largest prime factor prime(n), the largest exponent cannot exceed 2*log_2(prime(n+1)). (See formula 54 on page 15 of the Ramanujan paper.) This limit is less than n for all n >= 9 (and prime(n) >= 23).
1. Direct calculation verifies this for 9 <= n <= 11.
2. Nagura proved that, for any integer m >= 25, there is always a prime between m and 1.2*m.  Let n = 11, at which point prime(11) = 31 and log_2(prime(n+1)) = log 37/log 2 = 5.209453.... Since log 1.2/log 2 is only 0.263034..., it follows that n must increase by at least 3k before 2*log_2(prime(n+1)) can increase by 2k, for all values of k. Therefore, 2*log_2(prime(n+1)) can never catch up to prime(n) for n > 11.
665280 = 2^6*3^3*5*7*11 is the largest highly composite number whose prime factorization contains an exponent that is strictly greater than the number of positive exponents in that factorization (including the implied 1's).

			A002182(62) = 294053760 = 2^7*3^3*5*7*11*13*17 has 7 positive exponents in its prime factorization, including 5 implied 1's. The maximal exponent in its prime factorization is also 7. Therefore, 294053760 is a term of this sequence.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

A. Flammenkamp, List of the first 1200 highly composite numbers
J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.
S. Ramanujan, Highly Composite Numbers (p. 15) (pages 11-12 introduce some of the notation in formula 54)

A212165 also includes all terms in A006939, A066120, A087980, A130091, A138534, A141586, A166475, A181555, A181813-A181814, A181818, A181823-A181825, A182763.

Other finite subsequences of A002182 include A072938, A095921, A106037, A136253, A162935, A166981, A168263.

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; a = 0; t = {}; Do[b = DivisorSigma[0, n]; If[b > a, a = b; If[okQ[n], AppendTo[t, n]]], {n, 10^6}]; t (* T. D. Noe, May 24 2012 *)

A322488 Number of factorizations of the n-th power of the product of the first n primes.

Original entry on oeis.org

1, 1, 9, 686, 911838, 27119992809, 23970534519938280, 790361842548583118561351, 1186709456459520739315771458325336, 96786580459441954551347685958203256606168610, 502265575410823018475962732653887865889973989497971475955

Offset: 0

Alois P. Heinz, Dec 11 2018

nonn

Also number of n-partite partitions of {n}^n into n-tuples. a(2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].

Also number of partitions of the multiset with n copies each of 1,2,...,n. a(2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.

			a(2) = 9: (2*3)^2 = A002110(2)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.

Main diagonal of A219727.

Cf. A000040, A000110, A001055, A002110, A181555.

a(n) = A001055(A181555(n)) = A001055(A002110(n)^n).

a(10) from Andrew Howroyd, Dec 11 2018

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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A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

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A322488 Number of factorizations of the n-th power of the product of the first n primes.

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

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