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A355939
Dirichlet inverse of A080339, characteristic function of noncomposite numbers.
3
1, -1, -1, 1, -1, 2, -1, -1, 1, 2, -1, -3, -1, 2, 2, 1, -1, -3, -1, -3, 2, 2, -1, 4, 1, 2, -1, -3, -1, -6, -1, -1, 2, 2, 2, 6, -1, 2, 2, 4, -1, -6, -1, -3, -3, 2, -1, -5, 1, -3, 2, -3, -1, 4, 2, 4, 2, 2, -1, 12, -1, 2, -3, 1, 2, -6, -1, -3, 2, -6, -1, -10, -1, 2, -3, -3, 2, -6, -1, -5, 1, 2, -1, 12, 2, 2, 2, 4, -1, 12, 2, -3, 2, 2, 2, 6, -1, -3, -3, 6, -1, -6, -1, 4, -6
OFFSET
1,6
COMMENTS
The absolute values of this sequence are given by A008480. Compare also to A355817 and A335452.
FORMULA
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A010051(n/d) * a(d).
Dirichlet g.f.: 1/(1 + B(s)), where B(s) is d.g.f. of characteristic function of primes. - Vaclav Kotesovec, Jul 22 2022
MATHEMATICA
s[n_] := If[CompositeQ[n], 0, 1]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)
PROG
(PARI)
memoA355939 = Map();
A355939(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355939, n, &v), v, v = -sumdiv(n, d, if(d<n, isprime(n/d)*A355939(d), 0)); mapput(memoA355939, n, v); (v)));
CROSSREFS
KEYWORD
sign
AUTHOR
Antti Karttunen, Jul 21 2022
STATUS
approved