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A257463 Number A(n,k) of factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals. 9
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 3, 10, 17, 1, 1, 1, 1, 3, 23, 93, 73, 1, 1, 1, 1, 4, 40, 465, 1417, 388, 1, 1, 1, 1, 4, 73, 1746, 19834, 32152, 2461, 1, 1, 1, 1, 5, 114, 5741, 190131, 1532489, 1016489, 18155, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Also number of ways to partition the multiset consisting of k copies each of 1, 2, ..., n into n multisets of size k.

LINKS

Alois P. Heinz, Antidiagonals n = 0..12, flattened

P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. The array is on page 40.

Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets

Marko Riedel, Maple program to compute array from cycle indices

EXAMPLE

A(4,2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4.

A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.

A(2,4) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16.

Square array A(n,k) begins:

  1, 1,   1,     1,       1,        1,         1, ...

  1, 1,   1,     1,       1,        1,         1, ...

  1, 1,   2,     2,       3,        3,         4, ...

  1, 1,   5,    10,      23,       40,        73, ...

  1, 1,  17,    93,     465,     1746,      5741, ...

  1, 1,  73,  1417,   19834,   190131,   1398547, ...

  1, 1, 388, 32152, 1532489, 43816115, 848597563, ...

MAPLE

with(numtheory):

b:= proc(n, i, k) option remember; `if`(n=1, 1,

      add(`if`(d>i or bigomega(d)<>k, 0,

      b(n/d, d, k)), d=divisors(n)))

    end:

A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k$2, k):

seq(seq(A(n, d-n), n=0..d), d=0..8);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n==1, 1, DivisorSum[n, If[#>i || PrimeOmega[#] != k, 0, b[n/#, #, k]]&]];

A[n_, k_] := b[p = Product[Prime[i], {i, 1, n}]^k, p, k];

Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Mar 20 2017, translated from Maple *)

CROSSREFS

Columns k=0+1, 2-4 give: A000012, A002135, A254243, A268668.

Rows n=0+1, 2-5 give: A000012, A008619, A257464, A253259, A253263.

Cf. A257462, A257493 (ordered factorizations).

Sequence in context: A265313 A106498 A093466 * A293483 A125761 A154950

Adjacent sequences:  A257460 A257461 A257462 * A257464 A257465 A257466

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 24 2015

STATUS

approved

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Last modified January 21 21:30 EST 2020. Contains 331128 sequences. (Running on oeis4.)