A363900 -id:A363900 - cp's OEIS frontend

A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 4, 0, 2, 6, 0, 0, 6, 1, 0, 10, 2, 1, 8, 0, 0, 10, 4, 0, 12, 1, 0, 14, 0, 6, 12, 0, 2, 14, 0, 0, 20, 1, 4, 18, 2, 1, 16, 7, 0, 18, 0, 0, 20, 6, 8, 22, 0, 1, 24, 0, 2, 31, 0, 0, 28, 1, 0, 26, 12, 1, 24, 0, 0, 31, 4, 18, 28, 1, 0, 30, 0, 1

Offset: 1

N. J. A. Sloane, Dec 23 1999

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002131, A050460, A050469, A247037, A326400, A363900.

Cf. A050465, A050466, A050467.

a[n_] := DivisorSum[n, # &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

a(n)={sumdiv(n, d, d*(n/d%4==3))} \\ Andrew Howroyd, Sep 13 2019

G.f.: Sum_{k>=1} k*x^(3*k)/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 13 2019
G.f.: Sum_{k>0} x^(4*k-1) / (1 - x^(4*k-1))^2. - Seiichi Manyama, Jun 29 2023
from Amiram Eldar, Nov 05 2023: (Start)
a(n) = A002131(n) - A050460(n).
a(n) = A050460(n) - A050469(n).
a(n) = (A002131(n) - A050469(n))/2.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A247037. (End)

Offset changed to 1 by Ilya Gutkovskiy, Sep 13 2019

A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 5, 0, 7, 1, 6, 0, 8, 3, 10, 1, 9, 0, 15, 0, 13, 1, 15, 5, 14, 0, 16, 1, 21, 0, 21, 3, 19, 8, 18, 0, 20, 0, 35, 1, 24, 0, 27, 9, 25, 1, 30, 0, 36, 3, 28, 1, 27, 16, 40, 0, 31, 1, 45, 0, 32, 0, 42, 14, 39, 0, 39, 3, 56, 1, 45, 0, 38, 15, 40, 8, 42, 0, 71

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001822, A002131, A016789, A078182, A078708, A326399, A326401.

Cf. A050464, A363900.

nmax = 80; CoefficientList[Series[Sum[k x^(2 k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 80}]

a(n) = Sum_{d|n, n/d==2 (mod 3)} d.

G.f.: Sum_{k>0} x^(3*k-1) / (1 - x^(3*k-1))^2. - Seiichi Manyama, Jun 29 2023

A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 12, 14, 13, 14, 15, 17, 17, 21, 19, 20, 22, 24, 23, 28, 25, 27, 27, 28, 29, 35, 32, 34, 36, 34, 35, 43, 37, 38, 39, 40, 42, 51, 43, 48, 45, 47, 47, 59, 49, 50, 52, 54, 53, 63, 60, 57, 57, 58, 59, 70, 62, 64, 66, 68, 65, 84, 67, 68, 69, 70, 72, 86, 73, 74, 75, 77, 84, 94

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363898, A363899, A363900.
Cf. A002131, A050460, A326399.
Cf. A001876, A284097.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d%5==1)*d);
```

a(n) = Sum_{d|n, n/d==1 mod 5} d.

G.f.: Sum_{k>0} x^(5*k-4) / (1 - x^(5*k-4))^2.

A363898 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(5k)).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 7, 0, 9, 0, 8, 1, 9, 0, 10, 3, 12, 0, 14, 0, 13, 1, 18, 0, 15, 0, 17, 0, 19, 5, 21, 1, 19, 0, 20, 0, 28, 0, 24, 0, 23, 1, 28, 7, 25, 3, 27, 0, 29, 0, 36, 1, 29, 0, 35, 0, 32, 9, 34, 0, 36, 1, 38, 0, 45, 0, 43, 0, 39, 0, 38, 12, 39, 0, 40, 3, 42, 0, 63, 5, 43, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363897, A363899, A363900.

Cf. A001877, A284280.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d%5==2)*d);
```

a(n) = Sum_{d|n, n/d==2 mod 5} d.

G.f.: Sum_{k>0} x^(5*k-3) / (1 - x^(5*k-3))^2.

A363899 Expansion of Sum_{k>0} k * x^(3k) / (1 - x^(5k)).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 0, 4, 1, 0, 5, 2, 0, 7, 0, 0, 7, 0, 1, 11, 0, 2, 9, 1, 0, 10, 0, 4, 12, 0, 0, 14, 0, 1, 16, 5, 0, 14, 1, 0, 15, 2, 0, 23, 0, 0, 17, 4, 1, 21, 0, 9, 19, 1, 0, 20, 0, 0, 22, 8, 5, 24, 0, 1, 26, 0, 0, 37, 1, 0, 25, 2, 0, 33, 0, 10, 27, 0, 1, 31, 0, 2, 29, 12, 0

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A363897, A363898, A363900.

Cf. A001878, A284281.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 3 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d%5==3)*d);
```

a(n) = Sum_{d|n, n/d==3 mod 5} d.

G.f.: Sum_{k>0} x^(5*k-2) / (1 - x^(5*k-2))^2.

A364022 Expansion of Sum_{k>0} k * x^(4k) / (1 + x^(5k)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, 3, 0, 1, 0, 4, 0, -2, -1, 5, 0, 0, 0, 7, 0, 0, -3, 9, -1, 0, 0, 8, 0, 1, 0, 5, 0, -2, -1, 10, 0, 3, 0, 12, -5, 0, 0, 14, -1, 0, 0, 13, 0, -5, 0, 18, -3, -2, -1, 15, 0, 0, -7, 17, 0, 0, 0, 19, -1, 5, 0, 13, 0, 1, 0, 15, 0, -2, -1, 20, -9, 0, 0, 28, 0, 0, -3, 24, -1, -10, 0

Offset: 1

Seiichi Manyama, Jul 01 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A364019, A364020, A364021.

Cf. A363900.

a[n_] := DivisorSum[n, (-1)^(n/#) * # &, Mod[n/#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

a(n) = sumdiv(n, d, (n/d%5==4)*(-1)^(n/d)*d);

G.f.: Sum_{k>0} (-1)^(k-1) * x^(5*k-1) / (1 - x^(5*k-1))^2.

a(n) = Sum_{d|n, n/d==4 (mod 5)} (-1)^(n/d) * d.

cp's OEIS Frontend

A050464 a(n) = Sum_{d|n, n/d=3 mod 4} d.

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A326400 Expansion of Sum_{k>=1} k * x^(2*k) / (1 - x^(3*k)).

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A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

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A363898 Expansion of Sum_{k>0} k * x^(2*k) / (1 - x^(5*k)).

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A363899 Expansion of Sum_{k>0} k * x^(3*k) / (1 - x^(5*k)).

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A364022 Expansion of Sum_{k>0} k * x^(4*k) / (1 + x^(5*k)).

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PARI

Formula

A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

A363898 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(5k)).

A363899 Expansion of Sum_{k>0} k * x^(3k) / (1 - x^(5k)).

A364022 Expansion of Sum_{k>0} k * x^(4k) / (1 + x^(5k)).