A353348 Dirichlet inverse of A353350, where A353350 is the characteristic function for numbers k such that A048675(k) is a multiple of 3.

1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, -1, 0, 0

Offset: 1

Author

Antti Karttunen, Apr 15 2022

Keywords

• sign

Comments

From Antti Karttunen, Jan 17 2023: (Start)

a(120) = 3 is the first term with absolute value larger than 1.

As A353350 is not multiplicative, neither is this sequence.

For all numbers n in A359830 (complement of A332820), a(n) = 0. Proof: In the convolution formula, when n is any term of A359830, at least the other of the divisors (n/d) and d is a term of A359830, because a product of any two terms of A332820 stays in A332820 (as it is a multiplicative semigroup), while a product of a term of A332820 and its complement A359830 is always in A359830. As A353350 is zero for all A359830, it is easy to see by induction that also a(n) is zero for all such numbers. Therefore, nonzero values (including any odd values, see A359827) occur only on a subset of A332820, and A359826(n) <= A353350(n).

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A048675, A332820, A348717, A353349, A353350, A359826 (parity of terms), A359827 (positions of odd terms).

Cf. also A353418, A353421, A353422, A359773, A359780.

Programs

Mathematica

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

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(dA048675(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];

PARI

\\ Memoized variant, with no fixed limit for how many terms:
memoA353348 = Map();
A353348(n) = if(1==n,1,my(v); if(mapisdefined(memoA353348,n,&v), v, v = -sumdiv(n,d,if(dA353350(n/d)*A353348(d),0)); mapput(memoA353348,n,v); (v))); \\ Antti Karttunen, Jan 17 2023

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A353350(n/d) * a(d).

a(n) = A353349(n) - A353350(n).

a(p) = 0 for all primes p.

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

Extensions

Second offset added by Antti Karttunen, Jan 17 2023