A378526 -id:A378526 - cp's OEIS frontend

A378525 Dirichlet inverse of A378542, where A378542 is the sum of divisors d of n such that n/d has an even number of prime factors (counted with multiplicity).

1, -2, -3, -1, -5, 5, -7, 2, -1, 9, -11, 5, -13, 13, 14, 0, -17, 5, -19, 7, 20, 21, -23, -5, -1, 25, 3, 9, -29, -20, -31, 0, 32, 33, 34, -4, -37, 37, 38, -9, -41, -30, -43, 13, 8, 45, -47, -2, -1, 7, 50, 15, -53, -5, 54, -13, 56, 57, -59, -28, -61, 61, 10, 0, 64, -50, -67, 19, 68, -56, -71, -7, -73, 73, 8, 21, 76

Offset: 1

Antti Karttunen, Dec 01 2024

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

Cf. A378542.

Cf. also A378526.

A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
memoA378525 = Map();
A378525(n) = if(1==n,1,my(v); if(mapisdefined(memoA378525,n,&v), v, v = -sumdiv(n,d,if(dA378542(n/d)*A378525(d),0)); mapput(memoA378525,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378542(n/d) * a(d).

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).

A378527 Dirichlet inverse of A378546.

Original entry on oeis.org

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

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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).

cp's OEIS Frontend

A378525 Dirichlet inverse of A378542, where A378542 is the sum of divisors d of n such that n/d has an even number of prime factors (counted with multiplicity).

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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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A378527 Dirichlet inverse of A378546.

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