A363037 -id:A363037 - cp's OEIS frontend

A364011 Expansion of Sum_{k>0} x^k / (1 + x^(3*k)).

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

Offset: 1

Seiichi Manyama, Jul 01 2023

sign

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

Cf. A364012, A364013.

Cf. A002654, A363037.

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

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

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

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

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.

A364031 Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 8, 10, 8, 11, 12, 12, 14, 12, 16, 18, 20, 19, 16, 20, 22, 23, 24, 21, 24, 30, 28, 28, 24, 31, 32, 34, 36, 28, 40, 36, 38, 36, 32, 42, 40, 43, 44, 40, 46, 47, 48, 50, 42, 54, 48, 52, 60, 44, 56, 58, 56, 59, 48, 60, 62, 67, 64, 48, 68, 67, 72, 68, 56, 71, 80, 74, 72, 63, 76, 76, 72, 79

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

Cf. A050469, A364012, A364019.

Cf. A363037.

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

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

G.f.: Sum_{k>0} (-1)^(k-1) * x^(4*k-3) / (1 - x^(4*k-3))^2.

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

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

A364011 Expansion of Sum_{k>0} x^k / (1 + x^(3*k)).

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

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A364031 Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).

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

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