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

1,225

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

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A356163(n/d) * a(d).

Prog

(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(d<n, A356163(n/d)*A359773(d), 0)); mapput(memoA359773, n, v); (v)));

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.

Sequence in context: A010057 A359548 A359549 * A359774 A204220 A281814

Adjacent sequences: A359770 A359771 A359772 * A359774 A359775 A359776

Keyword

sign

Author

Antti Karttunen, Jan 13 2023

Status

approved