A307776 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+1)*a(d).

1, 1, 0, 1, -1, 0, 0, 1, -2, -1, 0, 1, 0, 1, 1, 1, -3, -2, 0, 1, 2, 3, 3, 4, 1, 1, 1, 0, -2, -1, -2, -1, -6, -5, -2, -2, 2, 3, 3, 4, 3, 4, 3, 4, 0, -1, -4, -3, -11, -10, -11, -13, -15, -14, -15, -15, -18, -17, -15, -14, -11, -10, -8, -7, -11, -11, -2, -1, 6, 10, 13, 14, 21, 22, 20

Offset: 1

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

• sign

Crossrefs

Cf. A003238, A281487, A307777, A307778, A307779.

Programs

Maple

with(numtheory): P:=proc(q) local a,d,n; a:=[1]:
for n from 1 to q do a:=[op(a),add((-1)^(n/d+1)*a[d],d=divisors(n))]:
od; op(a); end: P(74); # Paolo P. Lava, Apr 30 2019

Mathematica

a[n_] := a[n] = Sum[(-1)^((n - 1)/d + 1) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 75}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[a[k] x^k/(1 + x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 75}]

Formula

G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 + x^n)).

L.g.f.: log(Product_{n>=1} (1 + x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.