A364022 -id:A364022 - cp's OEIS frontend

A364019 Expansion of Sum_{k>0} k * x^k / (1 + x^(5*k)).

1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 12, 10, 13, 14, 15, 15, 17, 15, 19, 20, 22, 24, 23, 20, 25, 25, 27, 28, 29, 25, 32, 30, 36, 34, 35, 29, 37, 38, 39, 40, 42, 37, 43, 48, 45, 45, 47, 37, 49, 50, 52, 50, 53, 45, 60, 55, 57, 58, 59, 50, 62, 64, 66, 60, 65, 60, 67, 68, 69, 70, 72, 58, 73, 74, 75, 75, 84, 62, 79, 75

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

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

Cf. A364020, A364021, A364022.

Cf. A363897.

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

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

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

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

A364020 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, 5, 0, 8, -1, 9, 0, 10, -3, 12, 0, 14, 0, 13, -1, 10, 0, 15, 0, 17, 0, 15, -5, 21, -1, 19, 0, 20, 0, 16, 0, 24, 0, 23, -1, 28, -7, 25, -3, 27, 0, 25, 0, 20, -1, 29, 0, 35, 0, 32, -9, 34, 0, 36, -1, 30, 0, 25, 0, 43, 0, 35, 0, 38, -12, 39, 0, 40, -3, 42, 0, 39, -5, 43, -1

Offset: 1

Seiichi Manyama, Jul 01 2023

sign

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

Cf. A364019, A364021, A364022.

Cf. A363898.

a[n_] := DivisorSum[n, (-1)^(n/#) * # &, Mod[n/#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

a(n) = sumdiv(n, d, (n/d%5==2)*(-1)^(n/d)*d);

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

a(n) = Sum_{d|n, n/d==2 (mod 5)} (-1)^(n/d) * d.

A364021 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, 5, 0, 0, 7, 0, 1, 5, 0, 2, 9, -1, 0, 10, 0, -4, 12, 0, 0, 10, 0, -1, 16, -5, 0, 14, 1, 0, 15, 2, 0, 9, 0, 0, 17, 4, 1, 15, 0, -9, 19, -1, 0, 20, 0, 0, 22, -8, 5, 24, 0, -1, 26, 0, 0, 11, 1, 0, 25, -2, 0, 31, 0, -10, 27, 0, 1, 25, 0, 2, 29, -12, 0, 25

Offset: 1

Seiichi Manyama, Jul 01 2023

sign

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

Cf. A364019, A364020, A364022.

Cf. A363899.

a[n_] := -DivisorSum[n, (-1)^(n/#) * # &, Mod[n/#, 5] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

a(n) = -sumdiv(n, d, (n/d%5==3)*(-1)^(n/d)*d);

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

a(n) = -Sum_{d|n, n/d==3 (mod 5)} (-1)^(n/d) * d.

A364046 Expansion of Sum_{k>0} x^(4k) / (1 + x^(5k)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 1, 0, 1, 0, 1, 0, -1, -1, 1, 0, 0, 0, 2, 0, 0, -1, 2, -1, 0, 0, 1, 0, 1, 0, 0, 0, -1, -1, 1, 0, 1, 0, 2, -1, 0, 0, 2, -1, 0, 0, 1, 0, 0, 0, 2, -1, -1, -1, 1, 0, 0, -1, 2, 0, 0, 0, 2, -1, 1, 0, 1, 0, 1, 0, 0, 0, -1, -1, 1, -1, 0, 0, 3, 0, 0, -1, 2, -1, -1, 0, 1, 0, 1, -1, 2, 0, 0, -2

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A364043, A364044, A364045.

Cf. A001899, A364022.

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

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

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

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

cp's OEIS Frontend

A364019 Expansion of Sum_{k>0} k * x^k / (1 + x^(5*k)).

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A364020 Expansion of Sum_{k>0} k * x^(2k) / (1 + x^(5k)).

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A364021 Expansion of Sum_{k>0} k * x^(3k) / (1 + x^(5k)).

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A364046 Expansion of Sum_{k>0} x^(4k) / (1 + x^(5k)).

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cp's OEIS Frontend

A364019 Expansion of Sum_{k>0} k * x^k / (1 + x^(5*k)).

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A364020 Expansion of Sum_{k>0} k * x^(2*k) / (1 + x^(5*k)).

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A364021 Expansion of Sum_{k>0} k * x^(3*k) / (1 + x^(5*k)).

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A364046 Expansion of Sum_{k>0} x^(4*k) / (1 + x^(5*k)).

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A364020 Expansion of Sum_{k>0} k * x^(2k) / (1 + x^(5k)).

A364021 Expansion of Sum_{k>0} k * x^(3k) / (1 + x^(5k)).

A364046 Expansion of Sum_{k>0} x^(4k) / (1 + x^(5k)).