A256384 Number A(n,k) of factorizations of m^n into at most n factors, where m is a product of exactly k distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 19, 5, 1, 1, 1, 41, 171, 74, 7, 1, 1, 1, 122, 1675, 1975, 248, 11, 1, 1, 1, 365, 16683, 64182, 20096, 814, 15, 1, 1, 1, 1094, 166699, 2203215, 2213016, 187921, 2457, 22, 1, 1, 1, 3281, 1666731, 76727374, 268446852, 69406700, 1609727, 7168, 30, 1

Offset: 0

Author

Alois P. Heinz, Mar 27 2015

Keywords

• nonn
• tabl

Comments

A(n,k) is also the number of k-partite partitions of (n)^k into at most n k-tuples. A(2,2) = 5: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(1,2),(1,0)], [(1,1),(1,1)].

Examples

			A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into at most 2 factors: 36, 2*18, 3*12, 4*9, 6*6.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,         1, ...
  1, 1,   1,     1,       1,         1, ...
  1, 2,   5,    14,      41,       122, ...
  1, 3,  19,   171,    1675,     16683, ...
  1, 5,  74,  1975,   64182,   2203215, ...
  1, 7, 248, 20096, 2213016, 268446852, ...
		

Crossrefs

Columns k=0-3 give: A000012, A000041, A254686, A254811.

Rows n=0+1,2-3 give: A000012, A007051, A256493.

Cf. A219727.

Programs

Mathematica

b[n_, k_, i_] := b[n, k, i] = If[n>k, 0, 1] + If[PrimeQ[n] || i<2, 0, Sum[ If[d > k, 0, b[n/d, d, i-1]], {d, Divisors[n][[2 ;; -2]]}]]; A[0, ] = 1; A[1, ] = 1; A[, 0] = 1; A[n, k_] := With[{t = Times @@ Prime[ Range[k] ]}, b[t^n, t^n, n]]; Table[diag = Table[A[n-k, k], {k, n, 0, -1}]; Print[ diag]; diag, {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)