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A257462 Number A(n,k) of factorizations of m^n into n factors, where m is a product of exactly k 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. 5
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 10, 10, 3, 1, 1, 1, 1, 26, 70, 25, 3, 1, 1, 1, 1, 71, 566, 465, 49, 4, 1, 1, 1, 1, 197, 4781, 11131, 2505, 103, 4, 1, 1, 1, 1, 554, 41357, 297381, 190131, 12652, 184, 5, 1, 1, 1, 1, 1570, 364470, 8349223, 16669641, 2928876, 57232, 331, 5, 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 n copies each of 1, 2, ..., k into n multisets of size k.

LINKS

Table of n, a(n) for n=0..77.

EXAMPLE

A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*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) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.

Square array A(n,k) begins:

  1, 1, 1,  1,    1,      1, ...

  1, 1, 1,  1,    1,      1, ...

  1, 1, 2,  4,   10,     26, ...

  1, 1, 2, 10,   70,    566, ...

  1, 1, 3, 25,  465,  11131, ...

  1, 1, 3, 49, 2505, 190131, ...

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) minus {1}))

    end:

A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$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, Sum[If[d > i || PrimeOmega[d] != k, 0, b[n/d, d, k]], {d, Divisors[n] // Rest}]]; A[n_, k_] := Module[ {p = Product[Prime[i], {i, 1, k}]^n}, b[p, p, k]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-Fran├žois Alcover, Feb 22 2016, after Alois P. Heinz *)

CROSSREFS

Columns k=0+1, 2-5 give: A000012, A008619, A254233, A257114, A257518.

Rows n=0+1, 2 give: A000012, A257520.

Cf. A257463.

Sequence in context: A060097 A098120 A098873 * A046876 A026584 A247342

Adjacent sequences:  A257459 A257460 A257461 * A257463 A257464 A257465

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 24 2015

STATUS

approved

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Last modified October 20 17:50 EDT 2019. Contains 328268 sequences. (Running on oeis4.)