%I #10 Feb 16 2025 08:33:46

%S 1,2,3,4,5,7,8,10,11,13,14,17,18,20,22,24,25,28,29,32,34,36,37,42,43,

%T 45,47,50,51,56,57,60,62,64,66,71,72,74,76,81,82,87,88,91,94,96,97,

%U 104,105,108,110,113,114,119,121,126,128,130,131,140,141,143,146,150,152,157,158,161,163,168,169,178,179,181,184

%N Total number of unordered factorizations of all positive integers <= n into distinct factors greater than 1.

%C Partial sums of A045778.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnorderedFactorization.html">Unordered Factorization</a>

%F a(p^k) = a(p^k-1) + A000009(k), where p is a prime.

%e a(6) = 7 because we have [1], [2], [3], [4], [5], [2*3] and [6] (the factorization [2*2] is not permitted because the factor 2 is present twice).

%t Accumulate[gd[m_, 1] := 1; gd[1, n_] := 0; gd[1, 1] := 1; gd[0, n_] := 0; gd[m_, n_] := gd[m, n] = Total[gd[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; Array[ gd[#, #] &, 75]]

%Y Cf. A000009, A001055, A036469, A045778, A086435, A096276.

%K nonn,changed

%O 1,2

%A _Ilya Gutkovskiy_, May 26 2017