A355824 Dirichlet inverse of A355823, characteristic function of exponentially 2^n-numbers.

1, -1, -1, 0, -1, 1, -1, 1, 0, 1, -1, 0, -1, 1, 1, -2, -1, 0, -1, 0, 1, 1, -1, -1, 0, 1, 1, 0, -1, -1, -1, 2, 1, 1, 1, 0, -1, 1, 1, -1, -1, -1, -1, 0, 0, 1, -1, 2, 0, 0, 1, 0, -1, -1, 1, -1, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 2, -2, 1, -1, 0, 1, 1, 1, -1, -1, 0, 1, 0, 1, 1, 1, -2, -1, 0, 0, 0, -1, -1, -1, -1, -1, 1, -1, 0, -1, -1, 1, 2, -1, -1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, -1, -3

Offset: 1

Author

Antti Karttunen, Jul 19 2022

Keywords

• sign
• mult

Comments

Multiplicative because A355823 is.

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. A138302, A355823.

Differs from related A355826 for the first time at n=128, where a(128) = -3, while A355826(128) = -4.

Programs

Mathematica

s[n_] := If[AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 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

A355823(n) = factorback(apply(e->!bitand(e,e-1),factor(n)[,2]));
memoA355824 = Map();
A355824(n) = if(1==n,1,my(v); if(mapisdefined(memoA355824,n,&v), v, v = -sumdiv(n,d,if(dA355823(n/d)*A355824(d),0)); mapput(memoA355824,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA355823(n/d) * a(d).