A359608 -id:A359608 - cp's OEIS frontend

A046337 Odd numbers with an even number of prime factors (counted with multiplicity).

1, 9, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 81, 85, 87, 91, 93, 95, 111, 115, 119, 121, 123, 129, 133, 135, 141, 143, 145, 155, 159, 161, 169, 177, 183, 185, 187, 189, 201, 203, 205, 209, 213, 215, 217, 219, 221, 225, 235, 237, 247, 249, 253, 259

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A028260.
Setwise difference A005408 \ A067019.
Setwise difference A028260 \ A063745.
Union of A359161 and A359163.
Union of A327862 and A360110.
Subsequence of A345452, of A356312 and of A359371.
Positions of positive terms in A166698, positions of even terms in A327858 and A356299.
Subsequences: A002557, A046315 (odd semiprimes), A056913, A359596, A359607, A359608 (without its term 2).
Cf. A000035, A008836, A046338, A046470, A353557 (characteristic function), A358777.
Cf. also A036349, A297845.

Select[Range[1,301,2],EvenQ[PrimeOmega[#]]&] (* Harvey P. Dale, Jul 25 2011 *)

lista(nn) = {forstep(n=1, nn, 2, if (bigomega(n) % 2 == 0, print1(n, ", ")));} \\ Michel Marcus, Jul 04 2015

{k | A000035(k) > 0 and A008836(k) > 0}. - Antti Karttunen, Jan 13 2023

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

Original entry on oeis.org

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

Antti Karttunen, Dec 20 2022

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)

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

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.

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)));

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)

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A046337 Odd numbers with an even number of prime factors (counted with multiplicity).

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

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A359598 Indices of terms with record absolute values in A358777.

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A359609 Distinct values of A358777 in the order of their appearance.

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