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Dirichlet inverse of A353380.
1

%I #8 Jul 15 2022 09:54:40

%S 1,0,0,-1,0,-1,0,-1,-1,0,0,-1,0,-1,-1,1,0,-1,0,-1,0,0,0,2,-1,-1,-1,-1,

%T 0,0,0,1,-1,0,-1,2,0,-1,0,0,0,0,0,-1,-1,0,0,3,-1,-1,-1,-1,0,2,0,2,0,

%U -1,0,1,0,0,-1,-1,-1,0,0,-1,-1,0,0,5,0,-1,-1,-1,-1,0,0,2,1,0,0,1,0,-1,0,0,0,1,0,-1,-1,0,-1,-2,0,-1,-1,1,0,0,0,2,0

%N Dirichlet inverse of A353380.

%H Antti Karttunen, <a href="/A355685/b355685.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A353380(n/d) * a(d).

%F a(p) = 0 for all primes p.

%F a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

%o (PARI)

%o A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };

%o A353354(n) = sumdiv(n,d,A332823(d));

%o A353380(n) = (0==A353354(n));

%o memoA355685 = Map();

%o A355685(n) = if(1==n,1,my(v); if(mapisdefined(memoA355685,n,&v), v, v = -sumdiv(n,d,if(d<n,A353380(n/d)*A355685(d),0)); mapput(memoA355685,n,v); (v)));

%Y Cf. A003961, A048675, A332823, A348717, A353354, A353355, A353380.

%Y Cf. also A353348, A353418.

%K sign

%O 1,24

%A _Antti Karttunen_, Jul 14 2022