OFFSET
1,3
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) is multiplicative and a(2^e) = -1 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1+x^(2k-1)).
From Amiram Eldar, Jan 28 2024: (Start)
a(n) = (-1)^(n+1) * A117000(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)
MATHEMATICA
f[p_, e_] := If[1 < Mod[p, 8] < 7, ((-p)^(e+1)-1)/(-p-1), (p^(e+1)-1)/(p-1)]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)
PROG
(PARI) a(n)=if(n<1, 0, -sumdiv(n, d, d*(d%2)*(-1)^(n/d+(d+1)\4)))
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -1, p*=kronecker(2, p); (p^(e+1)-1)/(p-1)))))}
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Oct 29 2005
STATUS
approved