A095705 Triangular array read by rows: a(n, k) = sum of number of ordered factorizations of all prime signatures with n total prime factors and k distinct prime factors.

1, 2, 3, 4, 8, 13, 8, 46, 44, 75, 16, 124, 308, 308, 541, 32, 572, 1790, 2536, 2612, 4683, 64, 1568, 8352, 17028, 24704, 25988, 47293, 128, 6728, 40628, 137498, 187928, 277456, 296564, 545835, 256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261

Offset: 1

Author

Alford Arnold, Jul 04 2004, Nov 22 2005

Keywords

• nonn
• tabl

Comments

A093936 is an analogous array for unordered factorizations.

First column is A000079. First two diagonals are A000670 and A005649.

Examples

			There are two prime signatures with 5 total primes and 3 distinct primes: p^3*q*r and p^2*q^2*r. A074206(p^3*q*r) = 132 and A074206(p^2*q^2*r) = 176, so a(5, 3) = 132+176 = 308.
Array begins:
  1
  2, 3
  4, 8, 13
  8, 46, 44, 75
  16, 124, 308, 308, 541
  32, 572, 1790, 2536, 2612, 4683
  64, 1568, 8352, 17028, 24704, 25988, 47293
  128, 6728, 40628, 137498, 187928, 277456, 296564, 545835
  256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..210 (rows n=1..20, flattened)

Crossrefs

A035341 gives the row sums. Cf. A050324, A074206, A093936, A096443.

Programs

Mathematica

H[0] = 0; H[1] = 1; H[n_] := H[n] = Total[H /@ Most[Divisors[n]]];
T[n_, k_] := Module[{t = IntegerPartitions[n, {k}]}, Total[H /@ Times @@@ ((Prime[Range[k]]^#) & /@ t)]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 28 2025 *)

Extensions

Edited and extended by David Wasserman, Feb 22 2008