A318583 a(1) = a(2) = 1; for n > 2, a(n+2) = Sum_{d|n} mu(n/d)*a(d).

1, 1, 1, 0, 0, -1, -1, -2, -2, -2, -3, -2, -4, 0, -5, 1, -5, 3, -6, 7, -7, 10, -6, 13, -7, 17, -7, 21, -5, 22, -6, 31, -7, 30, -4, 35, -2, 33, -3, 39, 1, 34, 0, 42, -1, 33, 7, 39, 6, 23, 7, 32, 12, 16, 11, 18, 15, -1, 21, 4, 20, -27, 19, -21, 29, -52, 34, -56, 33, -85, 39, -80, 38, -130, 37

Offset: 1

Author

Ilya Gutkovskiy, Aug 29 2018

Keywords

• sign
• look

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A007439, A007554, A008683.

Programs

Maple

with(numtheory): P:=proc(q) local k,n,x; x:=[1,1]: for n from 3 to q do
x:=[op(x),add(mobius((n-2)/k)*x[k],k=divisors(n-2))]; od; op(x); end:
P(75); # Paolo P. Lava, May 15 2019

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = Sum[a[d] MoebiusMu[(n - 2)/d], {d, Divisors[n - 2]}]; Table[a[n], {n, 75}]

PARI

A318583(n) = if(n<=2,1,sumdiv(n-2,d,moebius((n-2)/d)*A318583(d))); \\ (A non-memoized implementation) - Antti Karttunen, Aug 29 2018

PARI

\\ A faster implementation:
up_to = 16384;
A318583list(up_to) = { my(u=vector(up_to)); u[1] = u[2] = 1; for(n=3, up_to, u[n] = sumdiv(n-2,d,moebius((n-2)/d)*u[d])); (u); };
v318583 = A318583list(up_to);
A318583(n) = v318583[n]; \\ Antti Karttunen, Aug 29 2018

Formula

G.f.: Sum_{n>=1} a(n+2)*x^n/(1 - x^n).

L.g.f.: -log(Product_{n>=1} (1 - x^n)^(a(n+2)/n)) = Sum_{n>=1} a(n)*x^n/n.