A287549 Total number of unordered factorizations of all positive integers <= n into distinct factors greater than 1.

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

Offset: 1

Author

Ilya Gutkovskiy, May 26 2017

Keywords

• nonn

Comments

Partial sums of A045778.

Examples

			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).
		

Links

• Eric Weisstein's World of Mathematics, Unordered Factorization

Crossrefs

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

Programs

Mathematica

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]]

Formula

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