A378542 -id:A378542 - 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

sign

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).

A378531 Dirichlet convolution of A378432 and A378542.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 2, 2, 3, 0, 4, 0, 3, 3, 6, 0, 4, 0, 4, 3, 3, 0, 14, 2, 3, 2, 4, 0, 6, 0, 10, 3, 3, 3, 18, 0, 3, 3, 14, 0, 6, 0, 4, 4, 3, 0, 30, 2, 4, 3, 4, 0, 14, 3, 14, 3, 3, 0, 30, 0, 3, 4, 22, 3, 6, 0, 4, 3, 6, 0, 48, 0, 3, 4, 4, 3, 6, 0, 30, 6, 3, 0, 30, 3, 3, 3, 14, 0, 30, 3, 4, 3, 3, 3, 74, 0, 4, 4, 18

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Möbius transform of A378533.

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

Cf. A008683, A378532 (Dirichlet inverse), A378432, A378533 (inverse Möbius transform), A378542.

Cf. also A345182.

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));
memoA378432 = Map();
A378432(n) = if(1==n,1,my(v); if(mapisdefined(memoA378432,n,&v), v, v = -sumdiv(n,d,if(dA296075(n/d)*A378432(d),0)); mapput(memoA378432,n,v); (v)));
A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
A378531(n) = sumdiv(n,d,A378432(d)*A378542(n/d));

a(n) = Sum_{d|n} A378432(d)*A378542(n/d).

a(n) = Sum_{d|n} A008683(d)*A378533(n/d).

A378533 Dirichlet convolution of A323910 and A378542.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 5, 3, 4, 1, 10, 1, 4, 4, 11, 1, 10, 1, 10, 4, 4, 1, 26, 3, 4, 5, 10, 1, 16, 1, 21, 4, 4, 4, 34, 1, 4, 4, 26, 1, 16, 1, 10, 10, 4, 1, 62, 3, 10, 4, 10, 1, 26, 4, 26, 4, 4, 1, 56, 1, 4, 10, 43, 4, 16, 1, 10, 4, 16, 1, 98, 1, 4, 10, 10, 4, 16, 1, 62, 11, 4, 1, 56, 4, 4, 4, 26, 1, 56, 4, 10, 4, 4, 4

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Inverse Möbius transform of A378531.

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

Cf. A323910, A378542.
Cf. A378531 (Möbius transform), A378534 (Dirichlet inverse).
Cf. also A378223.

A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
A033879(n) = (n+n-sigma(n));
memoA323910 = Map();
A323910(n) = if(1==n,1,my(v); if(mapisdefined(memoA323910,n,&v), v, v = -sumdiv(n,d,if(dA033879(n/d)*A323910(d),0)); mapput(memoA323910,n,v); (v)));
A378533(n) = sumdiv(n,d,A323910(d)*A378542(n/d));

a(n) = Sum_{d|n} A323910(d)*A378542(n/d).

a(n) = Sum_{d|n} A378531(d).

A378535 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).

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 6, 5, 7, 5, 10, 7, 12, 7, 9, 11, 16, 9, 18, 13, 13, 11, 22, 13, 21, 13, 20, 19, 28, 15, 30, 21, 21, 17, 25, 23, 36, 19, 25, 23, 40, 21, 42, 31, 30, 23, 46, 27, 43, 25, 33, 37, 52, 27, 41, 33, 37, 29, 58, 33, 60, 31, 44, 43, 49, 33, 66, 49, 45, 35, 70, 41, 72, 37, 46, 55, 61, 39, 78, 49, 61, 41, 82

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

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

Cf. A001222, A008683, A378535, A378536 (Dirichlet inverse).

A378542(n) = sumdiv(n,d,d*!(bigomega(n/d)%2));
A378535(n) = sumdiv(n,d,moebius(d)*A378542(n/d));

a(n) = Sum_{d|n} A008683(d)*A378542(n/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).

A378532 Dirichlet convolution of A296075 and A378525.

Original entry on oeis.org

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).

A378543 Sum of divisors d of n such that n/d has an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 5, 3, 7, 1, 11, 1, 9, 8, 10, 1, 16, 1, 15, 10, 13, 1, 25, 5, 15, 10, 19, 1, 32, 1, 21, 14, 19, 12, 35, 1, 21, 16, 35, 1, 42, 1, 27, 25, 25, 1, 51, 7, 36, 20, 31, 1, 50, 16, 45, 22, 31, 1, 72, 1, 33, 31, 42, 18, 62, 1, 39, 26, 60, 1, 80, 1, 39, 41, 43, 18, 72, 1, 71, 30, 43, 1, 94, 22, 45, 32, 65

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Agrees with A378549 on odd n.
Dirichlet convolution of A000027 with A066829.
Dirichlet convolution of A000010 (Euler phi) with A056924.

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

Cf. A000010, A000027, A000203, A056924, A066829, A378542.

Cf. also A378547, A378549.

A378543(n) = sumdiv(n,d,d*!!(bigomega(n/d)%2));

a(n) = Sum_{d|n} A066829(n/d)*d.
a(n) = Sum_{d|n} A000010(n/d)*A056924(d).
a(n) = A000203(n) - A378543(n).

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).

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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A378531 Dirichlet convolution of A378432 and A378542.

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A378533 Dirichlet convolution of A323910 and A378542.

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A378535 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).

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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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A378532 Dirichlet convolution of A296075 and A378525.

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A378534 Dirichlet convolution of A033879 and A378525.

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A378543 Sum of divisors d of n such that n/d has an odd number of prime factors (counted with multiplicity).

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A378536 Inverse Möbius transform of A378525.