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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.

%I A095705 #15 Jun 29 2025 04:13:58
%S A095705 1,2,3,4,8,13,8,46,44,75,16,124,308,308,541,32,572,1790,2536,2612,
%T A095705 4683,64,1568,8352,17028,24704,25988,47293,128,6728,40628,137498,
%U A095705 187928,277456,296564,545835,256,18768,228308,719056,1699184,2356560,3526448,3816548,7087261
%N 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.
%C A095705 A093936 is an analogous array for unordered factorizations.
%C A095705 First column is A000079. First two diagonals are A000670 and A005649.
%H A095705 Amiram Eldar, <a href="/A095705/b095705.txt">Table of n, a(n) for n = 1..210</a> (rows n=1..20, flattened)
%e A095705 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.
%e A095705 Array begins:
%e A095705   1
%e A095705   2, 3
%e A095705   4, 8, 13
%e A095705   8, 46, 44, 75
%e A095705   16, 124, 308, 308, 541
%e A095705   32, 572, 1790, 2536, 2612, 4683
%e A095705   64, 1568, 8352, 17028, 24704, 25988, 47293
%e A095705   128, 6728, 40628, 137498, 187928, 277456, 296564, 545835
%e A095705   256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261
%t A095705 H[0] = 0; H[1] = 1; H[n_] := H[n] = Total[H /@ Most[Divisors[n]]];
%t A095705 T[n_, k_] := Module[{t = IntegerPartitions[n, {k}]}, Total[H /@ Times @@@ ((Prime[Range[k]]^#) & /@ t)]];
%t A095705 Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jun 28 2025 *)
%Y A095705 A035341 gives the row sums. Cf. A050324, A074206, A093936, A096443.
%K A095705 nonn,tabl
%O A095705 1,2
%A A095705 _Alford Arnold_, Jul 04 2004, Nov 22 2005
%E A095705 Edited and extended by _David Wasserman_, Feb 22 2008