A355935 Dirichlet inverse of A091862, characteristic function of numbers for which A267116(n) = bigomega(n), where A267116 is the bitwise-OR of the exponents of primes in the prime factorization of n.

1, -1, -1, 0, -1, 2, -1, 0, 0, 2, -1, -2, -1, 2, 2, 0, -1, -2, -1, -2, 2, 2, -1, 2, 0, 2, 0, -2, -1, -6, -1, 0, 2, 2, 2, 6, -1, 2, 2, 2, -1, -6, -1, -2, -2, 2, -1, -2, 0, -2, 2, -2, -1, 2, 2, 2, 2, 2, -1, 10, -1, 2, -2, 0, 2, -6, -1, -2, 2, -6, -1, -8, -1, 2, -2, -2, 2, -6, -1, -2, 0, 2, -1, 10, 2, 2, 2, 2, -1, 10, 2, -2, 2, 2, 2, 2, -1, -2, -2, 6, -1, -6, -1, 2, -6

Offset: 1

Author

Antti Karttunen, Jul 21 2022

Keywords

• sign

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. A091862, A267116, A268375.

Programs

Mathematica

s[n_] := If[n == 1 || PrimeOmega[n] == BitOr @@ FactorInteger[n][[;; , 2]], 1, 0]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

PARI

A267116(n) = if(1==n, 0, fold(bitor, factor(n)[, 2]));
A091862(n) = (bigomega(n)==A267116(n));
memoA355935 = Map();
A355935(n) = if(1==n,1,my(v); if(mapisdefined(memoA355935,n,&v), v, v = -sumdiv(n,d,if(dA091862(n/d)*A355935(d),0)); mapput(memoA355935,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA091862(n/d) * a(d).