A364011 -id:A364011 - cp's OEIS frontend

A364012 Expansion of Sum_{k>0} k * x^k / (1 + x^(3*k)).

1, 2, 3, 3, 5, 6, 8, 6, 9, 9, 11, 9, 14, 16, 15, 11, 17, 18, 20, 13, 24, 21, 23, 18, 26, 28, 27, 24, 29, 27, 32, 22, 33, 33, 40, 27, 38, 40, 42, 25, 41, 48, 44, 31, 45, 45, 47, 33, 57, 47, 51, 42, 53, 54, 56, 48, 60, 57, 59, 39, 62, 64, 72, 43, 70, 63, 68, 49, 69, 72, 71, 54, 74, 76, 78

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

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

Cf. A364011, A364013.

Cf. A050469, A364019, A364031.

a[n_] := -DivisorSum[n, (-1)^(n/#) * # &, Mod[n/#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)

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

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

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

A363037 Expansion of Sum_{k>0} x^k / (1 + x^(4*k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

Cf. A002654, A364011.

Cf. A364031, A364032.

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

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

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

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

A364043 Expansion of Sum_{k>0} x^k / (1 + x^(5*k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A364044, A364045, A364046.
Cf. A002654, A363037, A364011, A364047.
Cf. A001876, A364019.

a[n_] := -DivisorSum[n, (-1)^# &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)

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

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

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

A364204 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 14 2023

sign

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

Cf. A001817, A048272, A364011, A364205, A364232.

nmax = 92; CoefficientList[Series[Sum[x^(3 k + 1)/(1 + x^(3 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) &, MemberQ[{1}, Mod[n/#, 3]] &], {n, 1, 92}]

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

A364013 Expansion of Sum_{k>0} k^2 * x^k / (1 + x^(3*k)).

Original entry on oeis.org

1, 4, 9, 15, 25, 36, 50, 60, 81, 99, 121, 135, 170, 200, 225, 239, 289, 324, 362, 371, 450, 483, 529, 540, 626, 680, 729, 750, 841, 891, 962, 956, 1089, 1155, 1250, 1215, 1370, 1448, 1530, 1483, 1681, 1800, 1850, 1811, 2025, 2115, 2209, 2151, 2451, 2479, 2601, 2550, 2809, 2916, 3026

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

Cf. A364011, A364012.

Cf. A050470.

a[n_] := -DivisorSum[n, (-1)^(n/#) * #^2 &, Mod[n/#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)

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

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

A364047 Expansion of Sum_{k>0} x^k / (1 + x^(6*k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 2, 0, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 2, 2, 2, 0, 0, 2, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 1, 2, 1, 1, 0, 2

Offset: 1

Seiichi Manyama, Jul 03 2023

nonn

Cf. A002654, A363037, A364011, A364043.

Cf. A279060.

a[n_] := DivisorSum[n, (-1)^((#-1)/6) &, Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

a(n) = sumdiv(n, d, (d%6==1)*(-1)^((d-1)/6));

G.f.: Sum_{k>0} (-1)^(k-1) * x^(6*k-5) / (1 - x^(6*k-5)).

a(n) = Sum_{d|n, d==1 (mod 6)} (-1)^((d-1)/6).

cp's OEIS Frontend

A364012 Expansion of Sum_{k>0} k * x^k / (1 + x^(3*k)).

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A363037 Expansion of Sum_{k>0} x^k / (1 + x^(4*k)).

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A364043 Expansion of Sum_{k>0} x^k / (1 + x^(5*k)).

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A364204 Expansion of Sum_{k>=0} x^(3*k+1) / (1 + x^(3*k+1)).

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A364013 Expansion of Sum_{k>0} k^2 * x^k / (1 + x^(3*k)).

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Formula

A364047 Expansion of Sum_{k>0} x^k / (1 + x^(6*k)).

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A364204 Expansion of Sum_{k>=0} x^(3k+1) / (1 + x^(3k+1)).