A359598 -id:A359598 - cp's OEIS frontend

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

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)

A359608 Indices at which A358777 attains a new value.

Original entry on oeis.org

1, 2, 9, 225, 315, 1155, 2835, 4725, 10395, 11025, 17325, 45045, 70875, 93555, 99225, 155925, 165375, 255255, 259875, 363825, 405405, 637875, 675675, 1063125, 1334025, 1488375, 1576575, 2297295, 2338875, 2480625, 3274425, 3472875, 3648645, 3828825, 5457375, 6081075, 10135125, 12006225, 12733875, 13395375

Offset: 1

Antti Karttunen, Jan 12 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..55

Apart from 2, a subsequence of A046337.
Cf. A353557, A358777, A359598 (subsequence), A359609 (the distinct values themselves).
Cf. also A359600.

For all n <> 2, A359600(a(n)) = a(n).

A359599 Terms of A358777 with record absolute values.

Original entry on oeis.org

1, 2, 3, 5, -6, -11, -18, -33, -61, 66, 126, 198, 241, 378, -522, 723, -762, -1017, -1176

Offset: 1

Antti Karttunen, Jan 12 2023

Cf. A358777, A359598.

Subsequence of A359609.

a(n) = A358777(A359598(n)).

A359597 Indices k such that A358777(k) is odd, and k is not an odd semiprime.

Original entry on oeis.org

1, 135, 189, 297, 315, 351, 375, 459, 495, 513, 525, 585, 621, 693, 735, 765, 783, 819, 825, 837, 855, 875, 975, 999, 1029, 1035, 1071, 1107, 1155, 1161, 1197, 1269, 1275, 1287, 1305, 1365, 1375, 1395, 1425, 1431, 1449, 1593, 1617, 1625, 1647, 1665, 1683, 1715, 1725, 1785, 1809, 1815, 1827, 1845, 1881

Offset: 1

Antti Karttunen, Jan 09 2023

nonn

2025 = 3^4 * 5^2, 30375 = 3^5 * 5^3 and 455625 == 3^6 * 5^4 are the first terms with respectively 6, 8 and 10 prime factors (with multiplicity).

Setwise difference A359596 \ A046315.
Odd terms of A065036 form a subset.
Cf. A353480, A359595, A359598.

isA359597(n) = ((!A353480(n)) && A359595(n));

cp's OEIS Frontend

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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A359608 Indices at which A358777 attains a new value.

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A359599 Terms of A358777 with record absolute values.

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A359597 Indices k such that A358777(k) is odd, and k is not an odd semiprime.

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