A317581 a(1) = 1; a(n > 1) = 1 + Sum_{d|n, d

1, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, -2, 0, 2, 2, 1, 0, -2, 0, -2, 2, 2, 0, 4, 1, 2, 0, -2, 0, -6, 0, 0, 2, 2, 2, 7, 0, 2, 2, 4, 0, -6, 0, -2, -2, 2, 0, -4, 1, -2, 2, -2, 0, 4, 2, 4, 2, 2, 0, 16, 0, 2, -2, 1, 2, -6, 0, -2, 2, -6, 0, -12, 0, 2, -2, -2, 2, -6, 0, -4

Offset: 1

Author

Gus Wiseman, Jul 31 2018

Keywords

• sign
• eigen

Comments

If p is prime, a(p^k) = 0 if k is odd, 1 if k is even. - Robert Israel, Aug 01 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001678, A002033, A007554, A038046, A067824.

Programs

Maple

f:= n -> 1 + add(numtheory:-mobius(n/d)*procname(d),d=numtheory:-divisors(n) minus {n}):
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 01 2018

Mathematica

a[n_]:=1+Sum[MoebiusMu[n/d]*a[d],{d,Most[Divisors[n]]}];
Array[a,100]

Python

from sympy import mobius, divisors
def A317581(n): return 1 + (0 if n == 1 else sum(mobius(n//d)*A317581(d) for d in divisors(n,generator=True) if d < n)) # Chai Wah Wu, Jan 14 2022