A378546 -id:A378546 - cp's OEIS frontend

A378527 Dirichlet inverse of A378546.

1, -2, -3, 0, -5, 6, -7, 0, -1, 10, -11, -1, -13, 14, 14, -1, -17, 2, -19, -1, 20, 22, -23, 2, -1, 26, 3, -1, -29, -28, -31, 2, 32, 34, 34, 3, -37, 38, 38, 2, -41, -40, -43, -1, 8, 46, -47, 3, -1, 2, 50, -1, -53, -6, 54, 2, 56, 58, -59, 8, -61, 62, 10, 0, 64, -64, -67, -1, 68, -68, -71, -6, -73, 74, 8, -1, 76, -76, -79, 5

Offset: 1

Antti Karttunen, Dec 01 2024

sign

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

Cf. A023900, A055615, A369974, A378546.

Cf. also A378525, A378526, A378528.

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));
memoA378527 = Map();
A378527(n) = if(1==n,1,my(v); if(mapisdefined(memoA378527,n,&v), v, v = -sumdiv(n,d,if(dA378546(n/d)*A378527(d),0)); mapput(memoA378527,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378546(n/d) * a(d).
a(n) = Sum_{d|n} A055615(d)*A369974(n/d).
a(n) = Sum_{d|n} A023900(d)*A378528(n/d).

A378542 Sum of divisors d of n such that n/d has an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 5, 7, 7, 10, 10, 11, 11, 17, 13, 15, 16, 21, 17, 23, 19, 27, 22, 23, 23, 35, 26, 27, 30, 37, 29, 40, 31, 42, 34, 35, 36, 56, 37, 39, 40, 55, 41, 54, 43, 57, 53, 47, 47, 73, 50, 57, 52, 67, 53, 70, 56, 75, 58, 59, 59, 96, 61, 63, 73, 85, 66, 82, 67, 87, 70, 84, 71, 115, 73, 75, 83, 97, 78, 96, 79, 115, 91

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Agrees with A378548 on odd n.
Dirichlet convolution of A000027 with A065043.
Dirichlet convolution of A000010 (Euler phi) with A038548.

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

Cf. A000010, A000027, A000203, A001222, A038548, A065043, A378525 (Dirichlet inverse), A378543.

Cf. also A378546, A378548.

A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));

a(n) = Sum_{d|n} A065043(n/d)*d.
a(n) = Sum_{d|n} A000010(n/d)*A038548(d).
a(n) = A000203(n) - A378543(n).

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)).

Original entry on oeis.org

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

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).

A378548 Sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 10, 11, 12, 13, 14, 16, 16, 17, 20, 19, 20, 22, 22, 23, 24, 26, 26, 30, 28, 29, 32, 31, 32, 34, 34, 36, 40, 37, 38, 40, 40, 41, 44, 43, 44, 53, 46, 47, 48, 50, 52, 52, 52, 53, 60, 56, 56, 58, 58, 59, 64, 61, 62, 73, 64, 66, 68, 67, 68, 70, 72, 71, 80, 73, 74, 83, 76, 78, 80, 79, 80, 91, 82

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Agrees with A378542 on odd n.
Dirichlet convolution of A000027 with A353557.
Dirichlet convolution of A000010 (Euler phi) with A369257.

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

Cf. A000010, A000203, A353557, A369257, A378526 (Dirichlet inverse), A378549.

Cf. also A002131, A378542, A378546.

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
A378548(n) = sumdiv(n,d,d*A353557(n/d));

a(n) = Sum_{d|n} A353557(n/d)*d.
a(n) = Sum_{d|n} A000010(n/d)*A369257(d).
a(n) = A000203(n) - A378549(n).

cp's OEIS Frontend

A378527 Dirichlet inverse of A378546.

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A378542 Sum of divisors d of n such that n/d has an even number of prime factors (counted with multiplicity).

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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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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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PARI

Formula

A378548 Sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).

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