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A353423 For even n, a(n) = -Sum_{d|n, d a(n/2) * a(d), and for odd n, a(n) = a(A064989(n)), with a(1) = 1.

%I A353423 #23 Apr 21 2022 17:22:59
%S A353423 1,-1,-1,0,-1,-1,-1,0,0,-1,-1,-2,-1,-1,-1,0,-1,0,-1,-2,-1,-1,-1,-8,0,
%T A353423 -1,0,-2,-1,-5,-1,0,-1,-1,-1,0,-1,-1,-1,-8,-1,-5,-1,-2,-2,-1,-1,-96,0,
%U A353423 0,-1,-2,-1,0,-1,-8,-1,-1,-1,-70,-1,-1,-2,0,-1,-5,-1,-2,-1,-5,-1,0,-1,-1,0,-2,-1,-5,-1,-96,0,-1,-1,-70
%N A353423 For even n, a(n) = -Sum_{d|n, d<n} a(n/2) * a(d), and for odd n, a(n) = a(A064989(n)), with a(1) = 1.
%C A353423 Apparently, for all i, j >= 1, A077462(i) = A077462(j) => a(i) = a(j).
%H A353423 Antti Karttunen, <a href="/A353423/b353423.txt">Table of n, a(n) for n = 1..10000</a>
%H A353423 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353423 a(p) = -1 for all primes p.
%F A353423 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%o A353423 (PARI)
%o A353423 A000265(n) = (n>>valuation(n,2));
%o A353423 A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
%o A353423 memoA353423 = Map();
%o A353423 A353423(n) = if(1==n,1,my(v); if(mapisdefined(memoA353423,n,&v), v, if(!(n%2), v = -sumdiv(n,d,if(d<n,A353423(n/2)*A353423(d),0)), v = A353423(A064989(n))); mapput(memoA353423,n,v); (v)));
%Y A353423 Cf. A003961, A064989, A077462, A348717.
%Y A353423 Cf. A070003 (positions of 0's), A167171 (positions of -1's), A096156 (positions of -2's), A007304 (positions of -5's), A086975 (positions of -70's), all these are so far conjectural. Also a subsequence of A178739 seems to give the positions of -96's.
%Y A353423 Cf. also A353454, A353457, A353458, A353467 for similar recurrences.
%K A353423 sign
%O A353423 1,12
%A A353423 _Antti Karttunen_, Apr 21 2022