A355817 Dirichlet inverse of A010055, characteristic function of powers of primes.

1, -1, -1, 0, -1, 2, -1, 0, 0, 2, -1, -1, -1, 2, 2, 0, -1, -1, -1, -1, 2, 2, -1, 0, 0, 2, 0, -1, -1, -6, -1, 0, 2, 2, 2, 2, -1, 2, 2, 0, -1, -6, -1, -1, -1, 2, -1, 0, 0, -1, 2, -1, -1, 0, 2, 0, 2, 2, -1, 6, -1, 2, -1, 0, 2, -6, -1, -1, 2, -6, -1, -1, -1, 2, -1, -1, 2, -6, -1, 0, 0, 2, -1, 6, 2, 2, 2, 0, -1, 6, 2

Offset: 1

Author

Antti Karttunen, Jul 19 2022

Keywords

• sign

Comments

Question: Are the absolute values of this sequence given by A335452? Compare also to A355939 and A008480.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A010055, A335452.

Cf. also A050363, A355827, A355939.

Programs

Mathematica

s[n_] := If[PrimeNu[n] < 2, 1, 0]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 19 2022 *)

PARI

A010055(n) = ((1==n)||isprimepower(n));
memoA355817 = Map();
A355817(n) = if(1==n,1,my(v); if(mapisdefined(memoA355817,n,&v), v, v = -sumdiv(n,d,if(dA010055(n/d)*A355817(d),0)); mapput(memoA355817,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA010055(n/d) * a(d).