A353349 - OEIS

A353349

Sum of A353350 and its Dirichlet inverse.

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 2, 0, 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, 0, 2, 0, 0, 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, 2

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OFFSET

1,1

COMMENTS

a(720) = -1 is the first negative term.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = A353348(n) + A353350(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A353348(d) * A353350(n/d).

a(p) = 0 for all primes p.

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

MATHEMATICA

f[p_, e_] := e*2^(PrimePi[p] - 1); s[1] = 1; s[n_] := Boole@Divisible[Plus @@ f @@@ FactorInteger[n], 3]; sinv[1] = 1; sinv[n_] := -DivisorSum[n, sinv[#]*s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* Amiram Eldar, Apr 15 2022 *)

PROG

(PARI)

up_to = 16384;

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

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

A353350(n) = (0==(A048675(n)%3));

v353348 = DirInverseCorrect(vector(up_to, n, A353350(n)));

A353348(n) = v353348[n];

A353349(n) = (A353348(n)+A353350(n));

CROSSREFS

Cf. A003961, A048675, A348717, A353348, A353350.

Sequence in context: A130706 A000038 A335462 * A349399 A228594 A281669

Adjacent sequences: A353346 A353347 A353348 * A353350 A353351 A353352

KEYWORD

sign

AUTHOR

Antti Karttunen, Apr 15 2022

STATUS

approved