A114592 Sum_{n>=1} a(n)/n^s = Product_{k>=2} (1 - 1/k^s).

1, -1, -1, -1, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 0, -1, 1, -1, 0, 1, 0, 0, 1, -1, 1, 0, 1, -1, 1, -1, 0, 1, 1, 0, 1

Offset

1,360

Comments

For n >= 2, Sum_{k|n} A001055(n/k) * a(k) = 0. A114591(n) = Sum_{k|n} a(k).

First entry greater than 1 in absolute value is a(360) = -2. - Gus Wiseman, Sep 15 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(1) = 1; for n>= 2, a(n) = sum, over ways to factor n into any number of distinct integers >= 2, of (-1)^(number of integers in a factorization). (See example.)

Example

24 can be factored into distinct integers (each >= 2) as 24; as 4*6, 3*8 and 2*12; and as 2*3*4. (A045778(24) = 5).
So a(24) = (-1)^1 + 3*(-1)^2 + (-1)^3 = 1, where the 1 exponent is due to the 1 factor of the 24 = 24 factorization and the 2 exponent is due to the 3 cases of 2 factors each of the 24 = 4*6 = 3*8 = 2*12 factorizations and the 3 exponent is due to the 24 = 2*3*4 factorization.

Mathematica

strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[Sum[(-1)^Length[f], {f, strfacs[n]}], {n, 100}] (* Gus Wiseman, Sep 15 2018 *)

Prog

(PARI)
A114592aux(n, k) = if(1==n, 1, sumdiv(n, d, if(d > 1 && d <= k && d < n, (-1)*A114592aux(n/d, d-1))) - (n<=k)); \\ After code in A045778.
A114592(n) = A114592aux(n, n); \\ Antti Karttunen, Jul 23 2017

Crossrefs

Cf. A001055, A045778, A114591.

Cf. A001222, A162247, A190938, A259936, A281116, A303386, A316441, A319237, A319238.

Sequence in context: A345950 A122895 A333248 * A344864 A140653 A342892

Adjacent sequences: A114589 A114590 A114591 * A114593 A114594 A114595

Keyword

sign

Author

Leroy Quet, Dec 11 2005

Extensions

More terms from Antti Karttunen, Jul 23 2017

Status

approved