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

1,2

Comments

A093936 is an analogous array for unordered factorizations.

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

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A238975 A098348 A131420 * A034776 A068791 A393412

Adjacent sequences: A095702 A095703 A095704 * A095706 A095707 A095708

Keyword

nonn,tabl

Author

Alford Arnold, Jul 04 2004, Nov 22 2005

Extensions

Edited and extended by David Wasserman, Feb 22 2008

Status

approved