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A353454 a(1) = 1, for n > 1, a(n) = -Sum_{d|n, d a(A064989(n)) * a(d).

%I A353454 #13 Apr 21 2022 17:23:18
%S A353454 1,-1,1,0,-1,1,1,0,0,1,-1,2,1,1,-1,0,-1,0,1,0,-3,1,-1,4,0,1,0,2,1,-1,
%T A353454 -1,0,-1,1,1,0,1,1,-3,0,-1,-1,1,0,0,1,-1,8,0,0,-1,2,1,0,-3,4,-3,1,-1,
%U A353454 -2,1,1,0,0,1,-1,-1,0,-1,3,1,0,-1,1,0,2,-1,-1,1,0,0,1,-1,-4,-3,1,-3,0,1,0,9,0,-1,1,1,16
%N A353454 a(1) = 1, for n > 1, a(n) = -Sum_{d|n, d<n} a(A064989(n)) * a(d).
%H A353454 Antti Karttunen, <a href="/A353454/b353454.txt">Table of n, a(n) for n = 1..10000</a>
%H A353454 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353454 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A353455(n) * a(d).
%F A353454 For all n >= 1, a(A000040(n)) = ((-1)^n).
%o A353454 (PARI)
%o A353454 A000265(n) = (n>>valuation(n,2));
%o A353454 A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
%o A353454 memoA353454 = Map();
%o A353454 A353454(n) = if(1==n,1,my(v); if(mapisdefined(memoA353454,n,&v), v, v = -sumdiv(n,d,if(d<n,A353454(A064989(n))*A353454(d),0)); mapput(memoA353454,n,v); (v)));
%Y A353454 Cf. A000040, A064989, A353455 [= a(A064989(n))].
%Y A353454 Cf. also A353423, A353457.
%K A353454 sign
%O A353454 1,12
%A A353454 _Antti Karttunen_, Apr 21 2022