A174273 -id:A174273 - cp's OEIS frontend

A378444 a(n) is the number of divisors d of n such that A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 3, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 4, 1, 1, 3, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 3, 2, 2, 2, 1, 3, 3, 1, 1, 4, 2, 1, 2, 2, 1, 3, 2, 2, 2, 1, 2, 3, 1, 2, 3, 3, 1, 2, 1, 2, 4

Offset: 1

Antti Karttunen, Nov 27 2024

nonn

Number of terms of A369002 that divide n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A378444, Sequence Machine.

Inverse Möbius transform of A369001.
Cf. A000005, A174273, A369002, A378445, A378546.
Cf. also A369257.

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369001(n) = !(A083345(n)%2);
A378444(n) = sumdiv(n,d,A369001(d));

a(n) = Sum_{d|n} A369001(d).
a(n) = A000005(n) - A378445(n).
a(n) = Sum_{d|n} A023900(d)*A378546(n/d).
a(n) = ceiling(A174273(n)/2). [Conjectured] - Antti Karttunen, May 14 2025

A378445 a(n) is the number of divisors d of n such that A083345(d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 1, 3, 1, 4, 1, 3, 2, 3, 1, 4, 1, 4, 2, 3, 1, 6, 1, 3, 2, 4, 1, 6, 1, 4, 2, 3, 2, 6, 1, 3, 2, 6, 1, 6, 1, 4, 3, 3, 1, 7, 1, 4, 2, 4, 1, 6, 2, 6, 2, 3, 1, 8, 1, 3, 3, 5, 2, 6, 1, 4, 2, 6, 1, 9, 1, 3, 3, 4, 2, 6, 1, 7, 2, 3, 1, 8, 2, 3, 2, 6, 1, 9, 2, 4, 2, 3, 2, 9, 1, 4, 3, 6, 1, 6, 1, 6, 4

Offset: 1

Antti Karttunen, Nov 27 2024

nonn

Number of terms of A369003 that divide n.

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

Inverse Möbius transform of A377874.
Cf. A000005, A369003, A378444.
Cf. also A174273, A378443.

A377874(n) = { my(f=factor(n)); (numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1])))%2); };
A378445(n) = sumdiv(n,d,A377874(d));

a(n) = Sum_{d|n} A377874(d).

a(n) = A000005(n) - A378444(n).

A378443 Inverse Möbius transform of A372573.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 3, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 4, 3, 3, 2, 2, 2, 3, 4, 4, 2, 2, 2, 4, 2, 2, 2, 3, 4, 4, 2, 2, 2, 4, 2, 4, 2, 2, 3, 2, 4, 4, 2, 4, 2, 2, 2, 4, 4, 2, 2, 4, 2, 4, 4, 2, 2, 2, 4, 4, 2, 3, 2, 3, 2, 4, 2, 4, 4

Offset: 1

Antti Karttunen, Nov 27 2024

nonn

Number of terms of A339746 that divide n.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A339746, A372573.

Cf. also A174273, A378444.

A372573(n) = !((valuation(n, 2)-valuation(n, 3))%3);
A378443(n) = sumdiv(n,d,A372573(d));

a(n) = Sum_{d|n} A372573(d).

cp's OEIS Frontend

A378444 a(n) is the number of divisors d of n such that A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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A378445 a(n) is the number of divisors d of n such that A083345(d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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A378443 Inverse Möbius transform of A372573.

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