A364085 -id:A364085 - cp's OEIS frontend

A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

1, 3, 4, 5, 8, 7, 8, 14, 10, 11, 18, 13, 17, 22, 16, 17, 26, 26, 20, 30, 22, 23, 42, 25, 30, 38, 28, 38, 42, 31, 32, 55, 44, 35, 50, 37, 38, 65, 50, 41, 63, 43, 56, 62, 46, 58, 66, 62, 50, 81, 52, 53, 100, 55, 56, 78, 58, 74, 94, 74, 68, 86, 80, 65, 90, 67, 82, 124, 70, 71, 98, 86, 92, 117, 76, 77

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000203, A007503, A064216, A336840.

Cf. A364066, A364085.

a[n_] := DivisorSum[2*n - 1, # + 1 &]/2; Array[a, 100] (* Amiram Eldar, Jul 04 2023*)

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

a(n) = (1/2) * Sum_{d | 2*n-1} (d+1) = A007503(2*n-1)/2.
G.f.: Sum_{k>0} x^k / (1 - x^(2*k-1))^2.
a(n) = A336840(A064216(n)). - Antti Karttunen, Nov 30 2024

A364066 Expansion of Sum_{k>0} k * x^k / (1 - x^(3*k-1)).

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 10, 8, 10, 10, 15, 14, 14, 14, 20, 16, 20, 18, 28, 20, 22, 24, 30, 24, 26, 30, 40, 28, 30, 30, 40, 34, 39, 34, 48, 36, 44, 38, 50, 46, 42, 44, 58, 44, 46, 46, 74, 52, 50, 50, 68, 54, 54, 62, 70, 56, 66, 58, 82, 60, 76, 64, 80, 64, 66, 66, 97, 78, 70, 74, 90, 74, 74, 80, 114, 76, 88, 78, 100

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

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

Cf. A364063, A364085.

Cf. A359211, A363890.

a[n_] := DivisorSum[3*n - 1, # + 1 &, Mod[#, 3] == 2 &]/3; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)

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

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

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

A363259 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(4k-1)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 5, 0, 5, 3, 6, 0, 8, 0, 8, 4, 11, 0, 11, 0, 11, 5, 12, 2, 14, 0, 17, 6, 15, 0, 19, 0, 17, 7, 18, 0, 24, 5, 20, 8, 21, 0, 23, 0, 25, 9, 29, 0, 29, 0, 26, 16, 27, 0, 29, 0, 35, 11, 32, 3, 32, 0, 32, 12, 33, 7, 46, 0, 35, 13, 39, 0, 40, 0, 38, 14, 47, 0, 41, 8, 41, 22, 42, 0, 49, 0

Offset: 1

Seiichi Manyama, Jul 08 2023

nonn

Cf. A364084, A364085.

Cf. A363392.

a[n_] := DivisorSum[4*n - 2, # + 1 &, Mod[#, 4] == 3 &]/4; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)

a(n) = sumdiv(4*n-2, d, (d%4==3)*(d+1))/4;

a(n) = (1/4) * Sum_{d | 4*n-2, d==3 (mod 4)} (d+1).

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

A363359 Sum of divisors of 4n-1 of form 4k+3.

Original entry on oeis.org

3, 7, 11, 18, 19, 23, 30, 31, 42, 42, 43, 47, 54, 66, 59, 73, 67, 71, 93, 79, 83, 90, 98, 114, 113, 103, 107, 114, 138, 126, 126, 127, 131, 180, 139, 154, 157, 151, 186, 162, 163, 167, 193, 217, 179, 186, 198, 191, 252, 199, 210, 233, 211, 258, 222, 223, 227, 252, 282, 239, 273, 266, 251, 324, 266, 263, 270

Offset: 1

Seiichi Manyama, Jul 08 2023

nonn

Cf. A050452, A078703, A364085.

a[n_] := DivisorSum[4*n - 1, # &, Mod[#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 08 2023 *)

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

a(n) = A050452(4*n-1).

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

cp's OEIS Frontend

A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

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A364066 Expansion of Sum_{k>0} k * x^k / (1 - x^(3*k-1)).

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A363259 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(4k-1)).

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A363359 Sum of divisors of 4n-1 of form 4k+3.

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cp's OEIS Frontend

A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

Views

Author

Keywords

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Mathematica

PARI

Formula

A364066 Expansion of Sum_{k>0} k * x^k / (1 - x^(3*k-1)).

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Author

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Mathematica

PARI

Formula

A363259 Expansion of Sum_{k>0} k * x^(2*k) / (1 - x^(4*k-1)).

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PARI

Formula

A363359 Sum of divisors of 4*n-1 of form 4*k+3.

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Mathematica

PARI

Formula

A363259 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(4k-1)).

A363359 Sum of divisors of 4n-1 of form 4k+3.