A359773 Dirichlet inverse of A356163, where A356163 is the characteristic function of the numbers with an even sum of prime factors (counted with multiplicity).

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

Offset: 1

Author

Antti Karttunen, Jan 13 2023

Keywords

• sign

Comments

a(225) = 2 is the first term with absolute value larger than 1.

As A356163 is not multiplicative, neither is this sequence.

For all numbers n with an odd number of odd prime factors (with mult.), a(n) = 0. Proof: Numbers with an odd number of odd prime factors is sequence A335657 (equal to numbers whose odd part is in A067019). In the convolution formula, when n is any term of A335657, either the divisor (n/d) or d (but not both) is also a term of A335657. As A356163 is zero for all A335657, it is easy to show by induction that also a(n) is zero for all such numbers.

Therefore, nonzero values (including any odd values, see A359775) occur only on a subset of A036349, and A359774(n) <= A356163(n).

Links

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

Crossrefs

Cf. A001414, A036347, A036348, A036349, A067019, A335657, A356163, A359774 (parity of terms), A359775 (positions of odd terms), A359776 (of even terms), A359777.

Cf. also A359155, A359763 [= a(A003961(n))], A359780.

Programs

PARI

A356163(n) = (1-(((n=factor(n))[, 1]~*n[, 2])%2)); \\ After code in A001414.
memoA359773 = Map();
A359773(n) = if(1==n,1,my(v); if(mapisdefined(memoA359773,n,&v), v, v = -sumdiv(n,d,if(dA356163(n/d)*A359773(d),0)); mapput(memoA359773,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA356163(n/d) * a(d).