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%I #11 Jul 20 2022 08:50:20
%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,-3
%N Dirichlet inverse of A355823, characteristic function of exponentially 2^n-numbers.
%C Multiplicative because A355823 is.
%H Antti Karttunen, <a href="/A355824/b355824.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} A355823(n/d) * a(d).
%t 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 *)
%o (PARI)
%o A355823(n) = factorback(apply(e->!bitand(e,e-1),factor(n)[,2]));
%o memoA355824 = Map();
%o A355824(n) = if(1==n,1,my(v); if(mapisdefined(memoA355824,n,&v), v, v = -sumdiv(n,d,if(d<n,A355823(n/d)*A355824(d),0)); mapput(memoA355824,n,v); (v)));
%Y Cf. A138302, A355823.
%Y Differs from related A355826 for the first time at n=128, where a(128) = -3, while A355826(128) = -4.
%K sign,mult
%O 1,16
%A _Antti Karttunen_, Jul 19 2022