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

0,9

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)].

Links

Table of n, a(n) for n=0..65.

Example

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, ...

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 *)

Crossrefs

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

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

Cf. A219727.

Sequence in context: A008550 A064094 A090182 * A111673 A121391 A375555

Adjacent sequences: A256381 A256382 A256383 * A256385 A256386 A256387

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 27 2015

Status

approved