A378445 -id:A378445 - 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

A378545 a(n) is the sum of those divisors d of n for which A083345(d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

Original entry on oeis.org

0, 2, 3, 6, 5, 11, 7, 14, 3, 17, 11, 15, 13, 23, 8, 14, 17, 29, 19, 21, 10, 35, 23, 47, 5, 41, 30, 27, 29, 56, 31, 46, 14, 53, 12, 69, 37, 59, 16, 69, 41, 74, 43, 39, 53, 71, 47, 95, 7, 67, 20, 45, 53, 110, 16, 91, 22, 89, 59, 120, 61, 95, 73, 110, 18, 110, 67, 57, 26, 108, 71, 173, 73, 113, 83, 63, 18, 128, 79, 149

Offset: 1

Antti Karttunen, Nov 29 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors.

Cf. A000203, A377874, A378544.

Cf. also A378445 (number of such divisors).

A377874(n) = { my(f=factor(n)); (numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1])))%2); };
A378545(n) = sumdiv(n,d,d*A377874(d));

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

a(n) = A000203(n) - A378544(n).

A378547 a(n) is the sum of the divisors d of n for which A083345(n/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, 3, 1, 6, 1, 7, 3, 8, 1, 15, 1, 10, 8, 14, 1, 19, 1, 21, 10, 14, 1, 34, 5, 16, 10, 27, 1, 40, 1, 29, 14, 20, 12, 48, 1, 22, 16, 48, 1, 52, 1, 39, 25, 26, 1, 69, 7, 41, 20, 45, 1, 60, 16, 62, 22, 32, 1, 96, 1, 34, 31, 59, 18, 76, 1, 57, 26, 72, 1, 109, 1, 40, 41, 63, 18, 88, 1, 97, 30, 44, 1, 126, 22, 46, 32

Offset: 1

Antti Karttunen, Nov 30 2024

nonn

Dirichlet convolution of A000027 with A377874.

Dirichlet convolution of A000010 (Euler phi) with A378445.

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

Cf. A000010, A000027, A000203, A083345, A377874, A378445, A378546.

Cf. also A378544, A378545.

A377874(n) = { my(f=factor(n)); (numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1])))%2); };
A378547(n) = sumdiv(n,d,d*A377874(n/d));

a(n) = Sum_{d|n} A377874(n/d)*d.
a(n) = Sum_{d|n} A000010(n/d)*A378445(d).
a(n) = A000203(n) - A378546(n).

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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A378545 a(n) is the sum of those divisors d of n for which A083345(d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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Formula

A378547 a(n) is the sum of the divisors d of n for which A083345(n/d) is odd, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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