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A353349 Sum of A353350 and its Dirichlet inverse. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 16:00 EST 2022. Contains 358644 sequences. (Running on oeis4.)