A378444 -id:A378444 - cp's OEIS frontend

A378528 Dirichlet inverse of A378444.

1, -1, -1, 0, -1, 1, -1, 0, -1, 1, -1, -1, -1, 1, 0, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -1, -1, 0, -1, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 0, -1, -1, 2, 1, -1, 1, -1, 1, 0, -1, -1, -1, 0, 1, 0, 1, -1, 2, -1, 1, 2, 0, 0, 0, -1, -1, 0, 0, -1, -1, -1, 1, 2, -1, 0, 0, -1, 1, 0, 1, -1, 2, 0, 1, 0, 1, -1, -2, 0, -1, 0, 1, 0, -1, -1, 1, 2, 1, -1, 0, -1, 1, 2

Offset: 1

Antti Karttunen, Dec 01 2024

sign

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

Cf. A000010, A008683, A378444, A378527.

Möbius transform of A369974.

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));
memoA378528 = Map();
A378528(n) = if(1==n,1,my(v); if(mapisdefined(memoA378528,n,&v), v, v = -sumdiv(n,d,if(dA378444(n/d)*A378528(d),0)); mapput(memoA378528,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378444(n/d) * a(d).
a(n) = Sum_{d|n} A000010(n/d)*A378527(d).
a(n) = Sum_{d|n} A008683(n/d)*A369974(d).

A378546 a(n) is the sum of the divisors d of n for which A083345(n/d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 10, 11, 13, 13, 14, 16, 17, 17, 20, 19, 21, 22, 22, 23, 26, 26, 26, 30, 29, 29, 32, 31, 34, 34, 34, 36, 43, 37, 38, 40, 42, 41, 44, 43, 45, 53, 46, 47, 55, 50, 52, 52, 53, 53, 60, 56, 58, 58, 58, 59, 72, 61, 62, 73, 68, 66, 68, 67, 69, 70, 72, 71, 86, 73, 74, 83, 77, 78, 80, 79, 89, 91

Offset: 1

Antti Karttunen, Nov 30 2024

nonn

Dirichlet convolution of A000027 with A369001.

Dirichlet convolution of A000010 (Euler phi) with A378444.

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

Cf. A000010, A000027, A000203, A083345, A369001, A378444, A378547.

Cf. also A378544, A378545.

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369001(n) = !(A083345(n)%2);
A378546(n) = sumdiv(n,d,d*A369001(n/d));

a(n) = Sum_{d|n} A369001(n/d)*d.
a(n) = Sum_{d|n} A000010(n/d)*A378444(d).
a(n) = A000203(n) - A378547(n).

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).

A378544 a(n) is the sum of those divisors d of n for which A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 13, 1, 1, 16, 17, 1, 10, 1, 21, 22, 1, 1, 13, 26, 1, 10, 29, 1, 16, 1, 17, 34, 1, 36, 22, 1, 1, 40, 21, 1, 22, 1, 45, 25, 1, 1, 29, 50, 26, 52, 53, 1, 10, 56, 29, 58, 1, 1, 48, 1, 1, 31, 17, 66, 34, 1, 69, 70, 36, 1, 22, 1, 1, 41, 77, 78, 40, 1, 37, 91, 1, 1, 62, 86, 1, 88, 45, 1, 25

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, A369001, A378545.

Cf. also A378444 (number of such divisors).

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369001(n) = !(A083345(n)%2);
A378544(n) = sumdiv(n,d,d*A369001(d));

a(n) = Sum_{d|n} d*A369001(d).

a(n) = A000203(n) - A378545(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

A378528 Dirichlet inverse of A378444.

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A378546 a(n) is the sum of the divisors d of n for which A083345(n/d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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Formula

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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PARI

Formula

A378544 a(n) is the sum of those divisors d of n for which A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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PARI

Formula

A378443 Inverse Möbius transform of A372573.

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