A355827 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA302777(n/d) * a(d).

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

Offset: 1

Author

Antti Karttunen, Jul 19 2022

Keywords

• sign

Comments

Dirichlet inverse of function f(1) = 1, f(n) = A302777(n) for n > 1, which is the characteristic function of the union of {1} and "Fermi-Dirac primes", A050376.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A050376, A302777.

Cf. also A355817.

Programs

Mathematica

s[n_] := If[n > 1 && Length[(f = FactorInteger[n])] == 1 && (e = f[[;; , 2]]) == 2^IntegerExponent[e, 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

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
memoA355827 = Map();
A355827(n) = if(1==n,1,my(v); if(mapisdefined(memoA355827,n,&v), v, v = -sumdiv(n,d,if(dA302777(n/d)*A355827(d),0)); mapput(memoA355827,n,v); (v)));