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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.
6
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
KEYWORD
nonn,tabl
AUTHOR
Alford Arnold, Jul 04 2004, Nov 22 2005
EXTENSIONS
Edited and extended by David Wasserman, Feb 22 2008
STATUS
approved