A355826 Dirichlet inverse of A355825, characteristic function of exponentially odious 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, -4

Offset: 1

Author

Antti Karttunen, Jul 19 2022

Keywords

• sign
• mult

Comments

Multiplicative because A355825 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. A270428, A355825.

Differs from related A355824 for the first time at n=128, where a(128) = -4, while A355824(128) = -3.

Cf. also A355819.

Programs

Mathematica

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

PARI

A355825(n) = factorback(apply(e->(hammingweight(e)%2),factor(n)[,2]));
memoA355826 = Map();
A355826(n) = if(1==n,1,my(v); if(mapisdefined(memoA355826,n,&v), v, v = -sumdiv(n,d,if(dA355825(n/d)*A355826(d),0)); mapput(memoA355826,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA355825(n/d) * a(d).