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A355826 Dirichlet inverse of A355825, characteristic function of exponentially odious numbers.

%I A355826 #12 Jul 20 2022 08:50:30
%S A355826 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 A355826 -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 A355826 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 A355826 Dirichlet inverse of A355825, characteristic function of exponentially odious numbers.
%C A355826 Multiplicative because A355825 is.
%H A355826 Antti Karttunen, <a href="/A355826/b355826.txt">Table of n, a(n) for n = 1..100000</a>
%H A355826 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A355826 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A355825(n/d) * a(d).
%t A355826 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 A355826 (PARI)
%o A355826 A355825(n) = factorback(apply(e->(hammingweight(e)%2),factor(n)[,2]));
%o A355826 memoA355826 = Map();
%o A355826 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 A355826 Cf. A270428, A355825.
%Y A355826 Differs from related A355824 for the first time at n=128, where a(128) = -4, while A355824(128) = -3.
%Y A355826 Cf. also A355819.
%K A355826 sign,mult
%O A355826 1,16
%A A355826 _Antti Karttunen_, Jul 19 2022