A358777 Dirichlet inverse of A353557, the characteristic function of odd numbers with an even number of prime factors (counted with multiplicity).

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

Offset: 1

Author

Antti Karttunen, Dec 20 2022

Keywords

• sign
• easy

Comments

As A353557 is not multiplicative, neither is this sequence.

Absolute values differ from A353557 for the first time at n=81, where a(81) = 0.

Absolute values differ from A353480 for the first time at n=1, and then at n=135.

The first value greater than 1 occurs as a(225) = 2. The first value less than -1 occurs as a(2835) = -2.

From Antti Karttunen, Jan 12 2023: (Start)

Few properties concerning this sequence:

(1) For all even numbers n, a(n) = 0. Proof: In the convolution formula, at least the other of the divisors (n/d) and d is always even, for any such divisor pair of an even n. As A353557 is zero for all even numbers, it is easy to show by induction that also a(n) is zero for all even n.

(2) For all numbers n with an odd number of prime factors (with multiplicity), a(n) = 0. Proof: In the convolution formula, either the divisor (n/d) or d (but not both) has an odd number of prime factors for any divisor pair d and (n/d) of any n in A026424. As A353557 is zero for all A026424, it is easy to show by induction that also a(n) is zero for all such numbers.

(3) Therefore, nonzero values occur only on indices that are a subset of A046337. (See A359607 for exceptions).

(4) For any two odd numbers x and y with the same prime signature (A046523(x) = A046523(y)), a(x) = a(y).

(5) a(A046315(n)) = -1.

(6) Apparently it also holds that for any n that is a square that is the 4th, 6th, 8th, ..., 2k-th power (k>=2) of some natural number > 1, a(n) is even.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Jon Maiga, Computer-generated formulas for A358777, Sequence Machine.

Crossrefs

Cf. A046315, A046337, A065043, A353557, A358778 (positions of positive terms), A359595 (parity of terms), A359596 (positions of odd terms), A359599 (terms with record absolute values), A359598 (their positions in this sequence), A359607, A359609 (distinct values in the order of their appearance), A359608 (their positions in this sequence).

Agrees paritywise with A359589 and A366265.

Cf. also A323239 (Dirichlet inverse of A166698(n) = A353557(n) - A353558(n)).

Cf. A359763, A359773, A359780, A359814, A359815 for similar sequences.

Programs

PARI

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
memoA358777 = Map();
A358777(n) = if(1==n,1,my(v); if(mapisdefined(memoA358777,n,&v), v, v = -sumdiv(n,d,if(dA353557(n/d)*A358777(d),0)); mapput(memoA358777,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA353557(n/d) * a(d).

From Antti Karttunen, Nov 22 2023: (Start)

Following identities (among others) are listed by Sequence Machine:

a(n) = o(n)*A359763(n) = o(n)*A359773(n) = o(n)*A359780(n) = o(n)*A359814(n) = o(n)*A359815(n), where o(n) = A000035(n), parity of n.

a(n) = A353557(n) * A359763(n) = A353557(n) * A359814(n).

a(n) = A065043(n) * A359773(n).

(End)