A316441 a(n) = Sum (-1)^k where the sum is over all factorizations of n into factors > 1 and k is the number of factors.

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

Offset: 1

Author

Gus Wiseman, Jul 03 2018

Keywords

• sign

Comments

First term greater than 1 in absolute value is a(256) = 2.

Examples

			The factorizations of 24 are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24); so a(24) = 1 - 2 + 3 - 1 = 1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A001222, A001055, A045778, A114592, A162247, A190938, A259936, A281116, A303386.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,200}]

PARI

A316441(n, m=n, k=0) = if(1==n, (-1)^k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A316441(n/d, d, k+1))); (s)); \\ Antti Karttunen, Sep 08 2018, after Michael B. Porter's code for A001055

Formula

Dirichlet g.f.: Product_{n > 1} 1/(1 + 1/n^s).

Extensions

Secondary offset added by Antti Karttunen, Sep 08 2018