OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} (-1)^k * k * A(x^k).
Dirichlet g.f.: 1 / (zeta(s-1) * (1 - 2^(2 - s))).
a(n) = Sum_{d|n} A327268(d).
Multiplicative with a(2^e) = 2^(2*e-1), and a(p^e) = -p if e=1 and 0 for e>1, for odd primes p. - Amiram Eldar, Dec 02 2020
MATHEMATICA
a[1] = 1; a[n_] := n Sum[If[d < n, (-1)^(n/d) a[d]/d, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]
terms = 60; A[_] = 0; Do[A[x_] = x + Sum[(-1)^k k A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest
f[p_, e_] := If[p == 2, p^(2*e - 1), -p*Boole[e == 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Ilya Gutkovskiy, Feb 24 2020
STATUS
approved