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A355819 Dirichlet inverse of A270419, denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

%I A355819 #9 Jul 20 2022 08:49:57
%S A355819 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,-2,0,
%T A355819 -1,-1,-1,-8,1,1,1,0,-1,1,1,1,-1,-1,-1,0,0,1,-1,-2,0,0,1,0,-1,2,1,1,1,
%U A355819 1,-1,0,-1,1,0,12,1,-1,-1,0,1,-1,-1,0,-1,1,0,0,1,-1,-1,-2,4,1,-1,0,1,1,1,1,-1,0,1,0,1,1,1,8
%N A355819 Dirichlet inverse of A270419, denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).
%C A355819 Multiplicative because A270419 is.
%H A355819 Antti Karttunen, <a href="/A355819/b355819.txt">Table of n, a(n) for n = 1..16383</a>
%H A355819 Antti Karttunen, <a href="/A355819/a355819.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%F A355819 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A270419(n/d) * a(d).
%o A355819 (PARI)
%o A355819 A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<<i)); \\ From A065620
%o A355819 A270419(n) = {n=factor(n); n[, 2]=apply(A065620, n[, 2]); denominator(factorback(n)); }; \\ From A270419
%o A355819 memoA355819 = Map();
%o A355819 A355819(n) = if(1==n,1,my(v); if(mapisdefined(memoA355819,n,&v), v, v = -sumdiv(n,d,if(d<n,A270419(n/d)*A355819(d),0)); mapput(memoA355819,n,v); (v)));
%Y A355819 Cf. A065620, A270419.
%Y A355819 Cf. also A355826.
%K A355819 sign,mult
%O A355819 1,16
%A A355819 _Antti Karttunen_, Jul 19 2022