A378525 -id:A378525 - cp's OEIS frontend

A378532 Dirichlet convolution of A296075 and A378525.

1, 0, 0, -2, 0, -3, 0, -2, -2, -3, 0, -4, 0, -3, -3, -2, 0, -4, 0, -4, -3, -3, 0, -2, -2, -3, -2, -4, 0, -6, 0, -2, -3, -3, -3, -1, 0, -3, -3, -2, 0, -6, 0, -4, -4, -3, 0, -2, -2, -4, -3, -4, 0, -2, -3, -2, -3, -3, 0, 0, 0, -3, -4, -2, -3, -6, 0, -4, -3, -6, 0, 0, 0, -3, -4, -4, -3, -6, 0, -2, -2, -3, 0, 0, -3, -3, -3, -2

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Inverse Möbius transform of A378534.

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

Cf. A033879, A296075, A378531 (Dirichlet inverse), A378534 (Möbius transform), A378525, A378542.

Cf. also A378218.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
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)));
A378532(n) = sumdiv(n,d,A296075(d)*A378525(n/d));

a(n) = Sum_{d|n} A296075(d)*A378525(n/d).

a(n) = Sum_{d|n} A378534(d).

A378534 Dirichlet convolution of A033879 and A378525.

Original entry on oeis.org

1, -1, -1, -2, -1, -2, -1, 0, -2, -2, -1, 1, -1, -2, -2, 0, -1, 1, -1, 1, -2, -2, -1, 2, -2, -2, 0, 1, -1, 2, -1, 0, -2, -2, -2, 4, -1, -2, -2, 2, -1, 2, -1, 1, 1, -2, -1, 0, -2, 1, -2, 1, -1, 2, -2, 2, -2, -2, -1, 6, -1, -2, 1, 0, -2, 2, -1, 1, -2, 2, -1, -1, -1, -2, 1, 1, -2, 2, -1, 0, 0, -2, -1, 6, -2, -2, -2, 2, -1, 6

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Möbius transform of A378532.

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

Cf. A008683, A033879, A323910, A378532 (inverse Möbius transform), A378533 (Dirichlet inverse), A378542.

Cf. also A378224.

A033879(n) = (n+n-sigma(n));
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)));
A378534(n) = sumdiv(n,d,A033879(d)*A378525(n/d));

a(n) = Sum_{d|n} A033879(d)*A378525(n/d).

a(n) = Sum_{d|n} A008683(d)*A378532(n/d).

A378536 Inverse Möbius transform of A378525.

Original entry on oeis.org

1, -1, -2, -2, -4, 1, -6, 0, -3, 3, -10, 5, -12, 5, 7, 0, -16, 5, -18, 9, 11, 9, -22, 2, -5, 11, 0, 13, -28, -1, -30, 0, 19, 15, 23, 5, -36, 17, 23, 2, -40, -3, -42, 21, 14, 21, -46, 0, -7, 9, 31, 25, -52, 3, 39, 2, 35, 27, -58, -18, -60, 29, 20, 0, 47, -7, -66, 33, 43, -13, -70, -5, -72, 35, 14, 37, 59, -9, -78, 0, 0

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Dirichlet inverse of A378535, which is Möbius transform 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).

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

Cf. A378525, A378535 (Dirichlet inverse), A378542.

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)));
A378536(n) = sumdiv(n,d,A378525(d));

a(n) = Sum_{d|n} A378525(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).

A378526 Dirichlet inverse of A378548, where A378548 is the 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, 0, -5, 6, -7, 0, -1, 10, -11, 0, -13, 14, 14, 0, -17, 2, -19, 0, 20, 22, -23, 0, -1, 26, 3, 0, -29, -28, -31, 0, 32, 34, 34, 0, -37, 38, 38, 0, -41, -40, -43, 0, 8, 46, -47, 0, -1, 2, 50, 0, -53, -6, 54, 0, 56, 58, -59, 0, -61, 62, 10, 0, 64, -64, -67, 0, 68, -68, -71, 0, -73, 74, 8, 0, 76, -76, -79, 0, 0, 82

Offset: 1

Antti Karttunen, Dec 01 2024

sign

Agrees with A378525 on all odd n, and also on some even n: 2, 16, 32, 64, 96, 128, 160, 192, ...

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

Cf. A023900, A055615, A353557, A358777, A369454, A378548.

Cf. also A378525, A378527.

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
A378548(n) = sumdiv(n,d,d*A353557(n/d));
memoA378526 = Map();
A378526(n) = if(1==n,1,my(v); if(mapisdefined(memoA378526,n,&v), v, v = -sumdiv(n,d,if(dA378548(n/d)*A378526(d),0)); mapput(memoA378526,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378548(n/d) * a(d).
a(n) = Sum_{d|n} A023900(d)*A369454(n/d).
a(n) = Sum_{d|n} A055615(d)*A358777(n/d).

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

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).

cp's OEIS Frontend

A378532 Dirichlet convolution of A296075 and A378525.

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A378534 Dirichlet convolution of A033879 and A378525.

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A378536 Inverse Möbius transform of A378525.

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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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A378526 Dirichlet inverse of A378548, where A378548 is the 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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