A363901 -id:A363901 - cp's OEIS frontend

A363903 Expansion of Sum_{k>0} x^k / (1 - x^(4*k))^2.

1, 1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 5, 1, 3, 1, 6, 4, 1, 3, 7, 1, 1, 1, 10, 5, 4, 1, 9, 3, 1, 1, 10, 6, 3, 4, 11, 1, 5, 3, 12, 7, 1, 1, 18, 1, 1, 1, 14, 10, 6, 5, 15, 4, 3, 1, 16, 9, 1, 3, 17, 1, 10, 1, 24, 10, 1, 6, 19, 3, 1, 4, 20, 11, 10, 1, 21, 5, 1, 3, 25, 12, 1, 7, 30, 1, 9, 1, 24, 18, 5, 1, 25, 1, 3, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A363901, A363925.

Cf. A001826, A050449.

a[n_] := DivisorSum[n, # + 3 &, Mod[#, 4] == 1 &]/4; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%4==1)*(d+3))/4;
```

a(n) = (1/4) * Sum_{d|n, d==1 mod 4} (d+3) = (3 * A001826(n) + A050449(n))/4.

G.f.: Sum_{k>0} k * x^(4*k-3) / (1 - x^(4*k-3)).

A363925 Expansion of Sum_{k>0} x^k / (1 - x^(5*k))^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 5, 1, 3, 1, 1, 6, 4, 1, 3, 1, 7, 1, 1, 1, 3, 8, 5, 4, 1, 1, 11, 1, 1, 1, 1, 10, 8, 1, 4, 1, 11, 1, 7, 1, 1, 12, 7, 1, 3, 4, 13, 1, 1, 1, 3, 14, 8, 6, 5, 1, 20, 1, 1, 1, 1, 16, 11, 1, 1, 1, 17, 4, 9, 1, 5, 18, 10, 1, 8, 1, 19, 1, 4, 1, 3, 20, 11

Offset: 1

Seiichi Manyama, Jun 28 2023

nonn

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

Cf. A363926, A363928, A363929.
Cf. A001876, A284097.
Cf. A363901, A363903.

a[n_] := DivisorSum[n, # + 4 &, Mod[#, 5] == 1 &] / 5; Array[a, 100] (* Amiram Eldar, Jun 28 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%5==1)*(d+4))/5;
```

a(n) = (1/5) * Sum_{d|n, d==1 mod 5} (d+4) = (4 * A001876(n) + A284097(n))/5.

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

A363902 Expansion of Sum_{k>0} x^(2k) / (1 - x^(3k))^2.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 4, 0, 3, 4, 1, 0, 6, 2, 4, 6, 1, 0, 10, 0, 5, 8, 4, 2, 10, 0, 6, 10, 3, 0, 15, 4, 7, 14, 1, 0, 14, 0, 13, 14, 6, 0, 20, 2, 9, 16, 4, 0, 20, 6, 10, 18, 1, 6, 28, 0, 11, 20, 10, 0, 22, 0, 15, 24, 5, 0, 30, 8, 20, 24, 4, 0, 26, 2, 14, 30, 10, 0, 40, 0, 15, 28, 6, 8

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A001822, A078182, A363901.

Cf. A113415, A363904.

a[n_] := DivisorSum[n, # + 1 &, Mod[#, 3] == 2 &]/3; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d%3==2)*(d+1))/3;
```

a(n) = (1/3) * Sum_{d|n, d==2 mod 3} (d+1) = (A001822(n) + A078182(n))/3.

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

A363970 Expansion of Sum_{k>0} k^2 * x^(3k-2) / (1 - x^(3k-2)).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 10, 5, 1, 17, 1, 5, 26, 10, 1, 41, 1, 1, 50, 21, 10, 65, 1, 5, 82, 26, 1, 114, 1, 17, 122, 41, 1, 145, 10, 5, 170, 50, 26, 217, 1, 10, 226, 69, 1, 257, 1, 41, 299, 98, 1, 354, 1, 1, 362, 114, 50, 401, 1, 21, 442, 122, 10, 525, 26, 65, 530, 149, 1, 602, 1, 5, 626, 170

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001817, A363901.

Cf. A103637, A363975.

a[n_] := DivisorSum[n, ((#+2)/3)^2 &, Mod[#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%3==1)*((d+2)/3)^2);

a(n) = Sum_{d|n, d==1 mod 3} ((d+2)/3)^2.

A363975 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^3.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 7, 4, 1, 11, 1, 4, 16, 7, 1, 25, 1, 1, 29, 14, 7, 37, 1, 4, 46, 16, 1, 65, 1, 11, 67, 25, 1, 79, 7, 4, 92, 29, 16, 119, 1, 7, 121, 40, 1, 137, 1, 25, 160, 56, 1, 190, 1, 1, 191, 65, 29, 211, 1, 14, 232, 67, 7, 278, 16, 37, 277, 82, 1, 317, 1, 4, 326, 92, 46, 383, 7, 16, 379

Offset: 1

Seiichi Manyama, Jun 30 2023

nonn

Cf. A001817, A363901.

Cf. A363970.

a[n_] := DivisorSum[n, Binomial[(#+2)/3+1,2] &, Mod[#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%3==1)*binomial((d+2)/3+1, 2));

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

a(n) = Sum_{d|n, d==1 mod 3} binomial((d+2)/3+1,2).

cp's OEIS Frontend

A363903 Expansion of Sum_{k>0} x^k / (1 - x^(4*k))^2.

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A363925 Expansion of Sum_{k>0} x^k / (1 - x^(5*k))^2.

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A363902 Expansion of Sum_{k>0} x^(2*k) / (1 - x^(3*k))^2.

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A363970 Expansion of Sum_{k>0} k^2 * x^(3*k-2) / (1 - x^(3*k-2)).

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A363975 Expansion of Sum_{k>0} x^k / (1 - x^(3*k))^3.

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A363902 Expansion of Sum_{k>0} x^(2k) / (1 - x^(3k))^2.

A363970 Expansion of Sum_{k>0} k^2 * x^(3k-2) / (1 - x^(3k-2)).