A355826 - OEIS

%I #12 Jul 20 2022 08:50:30

%S 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,

%T -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,

%U 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

%N Dirichlet inverse of A355825, characteristic function of exponentially odious numbers.

%C Multiplicative because A355825 is.

%H Antti Karttunen, <a href="/A355826/b355826.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A355825(n/d) * a(d).

%t 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 *)

%o (PARI)

%o A355825(n) = factorback(apply(e->(hammingweight(e)%2),factor(n)[,2]));

%o memoA355826 = Map();

%o A355826(n) = if(1==n,1,my(v); if(mapisdefined(memoA355826,n,&v), v, v = -sumdiv(n,d,if(d<n,A355825(n/d)*A355826(d),0)); mapput(memoA355826,n,v); (v)));

%Y Cf. A270428, A355825.

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

%Y Cf. also A355819.

%K sign,mult

%O 1,16

%A _Antti Karttunen_, Jul 19 2022