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

1,2

Comments

Partial sums of A045778.

Links

Table of n, a(n) for n=1..75.

Eric Weisstein's World of Mathematics, Unordered Factorization

Formula

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

Example

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).

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]]

Crossrefs

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

Sequence in context: A133223 A065003 A189729 * A347655 A056177 A025199

Adjacent sequences: A287546 A287547 A287548 * A287550 A287551 A287552

Keyword

nonn

Author

Ilya Gutkovskiy, May 26 2017

Status

approved