A353419 - OEIS

%I #8 Apr 19 2022 22:45:56

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

%T 0,2,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2

%N Sum of A353269 and its Dirichlet inverse.

%C The first negative term is a(840) = -2.

%H Antti Karttunen, <a href="/A353419/b353419.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(n) = A353269(n) + A353418(n).

%F For n > 1, a(n) = -Sum_{d|n, 1<d<n} A353269(d) * A353418(n/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 up_to = 65537;

%o DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v

%o A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };

%o A353269(n) = (!(A156552(n)%3));

%o v353418 = DirInverseCorrect(vector(up_to,n,A353269(n)));

%o A353418(n) = v353418[n];

%o A353419(n) = (A353269(n)+A353418(n));

%Y Cf. A003961, A156552, A348717, A353269, A353418.

%Y Cf. also A353349.

%K sign

%O 1,1

%A _Antti Karttunen_, Apr 19 2022