A359270 -id:A359270 - cp's OEIS frontend

A359288 Number of divisors of 5n-1 of form 5k+3.

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

Offset: 1

Seiichi Manyama, Dec 24 2022

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

Cf. A001878.
Cf. A359233, A359287.
Cf. A359236, A359244, A359270.

a[n_] := DivisorSum[5*n-1, 1 &, Mod[#, 5] == 3 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(5*n-1, d, d%5==3);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(3*k-1)/(1-x^(5*k-2)))))

a(n) = A001878(5*n-1).

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

A363035 Sum of divisors of 5n-3 of form 5k+3.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 8, 0, 3, 0, 13, 3, 0, 0, 29, 0, 0, 3, 23, 0, 3, 0, 36, 16, 0, 0, 36, 0, 0, 3, 46, 0, 21, 0, 43, 3, 13, 0, 59, 0, 0, 26, 53, 0, 3, 0, 66, 3, 0, 13, 112, 0, 0, 3, 76, 0, 3, 0, 73, 36, 0, 0, 102, 0, 23, 3, 83, 0, 59, 0, 96, 3, 0, 0, 96, 13, 0, 46, 134, 0, 3, 0, 103, 3, 0, 0, 185, 23, 13, 3, 113, 0, 36

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A363033, A363034, A363053.

Cf. A284281, A359270, A363157.

a[n_] := DivisorSum[5*n - 3, # &, Mod[#, 5] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 06 2023 *)

PARI
```
a(n) = sumdiv(5*n-3, d, (d%5==3)*d);
```

a(n) = A284281(5*n-3).

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

A364102 Sum of divisors of 5n-3 of form 5k+4.

Original entry on oeis.org

0, 0, 4, 0, 0, 9, 4, 0, 14, 0, 4, 19, 0, 0, 37, 0, 0, 29, 4, 0, 34, 0, 18, 48, 0, 0, 48, 0, 0, 49, 23, 0, 63, 0, 4, 59, 14, 0, 92, 0, 0, 78, 4, 0, 74, 0, 33, 79, 0, 19, 111, 0, 0, 89, 38, 0, 94, 0, 4, 108, 0, 0, 171, 0, 14, 109, 4, 0, 142, 0, 48, 119, 0, 0, 128, 29, 0, 138, 67, 0, 134, 0, 4, 139, 0, 0

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364100, A364101, A364103.

Cf. A284103, A359270, A364106.

a[n_] := DivisorSum[5*n - 3, # &, Mod[#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(5*n-3, d, (d%5==4)*d);
```

a(n) = A284103(5*n-3).

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

A364106 Expansion of Sum_{k>0} k * x^(3k) / (1 - x^(5k-1)).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 0, 3, 0, 1, 4, 0, 0, 8, 0, 0, 6, 1, 0, 7, 0, 4, 10, 0, 0, 10, 0, 0, 10, 5, 0, 13, 0, 1, 12, 3, 0, 19, 0, 0, 16, 1, 0, 15, 0, 7, 16, 0, 4, 23, 0, 0, 18, 8, 0, 19, 0, 1, 22, 0, 0, 35, 0, 3, 22, 1, 0, 29, 0, 10, 24, 0, 0, 26, 6, 0, 28, 14, 0, 27, 0, 1, 28, 0, 0, 48, 4, 7, 30, 1, 0

Offset: 1

Seiichi Manyama, Jul 05 2023

nonn

Cf. A364104, A364105, A364107.

Cf. A359270, A364102.

a[n_] := DivisorSum[5*n - 3, # + 1 &, Mod[#, 5] == 4 &]/5; Array[a, 100] (* Amiram Eldar, Jul 12 2023 *)

a(n) = sumdiv(5*n-3, d, (d%5==4)*(d+1))/5;

a(n) = (1/5) * Sum_{d | 5*n-3, d==4 (mod 5)} (d+1).

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

A363157 Expansion of Sum_{k>0} k * x^(4k-1) / (1 - x^(5k-2)).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 0, 0, 7, 0, 0, 1, 5, 0, 1, 0, 8, 4, 0, 0, 8, 0, 0, 1, 10, 0, 5, 0, 9, 1, 3, 0, 13, 0, 0, 6, 11, 0, 1, 0, 14, 1, 0, 3, 24, 0, 0, 1, 16, 0, 1, 0, 15, 8, 0, 0, 22, 0, 5, 1, 17, 0, 13, 0, 20, 1, 0, 0, 20, 3, 0, 10, 28, 0, 1, 0, 21, 1, 0, 0, 39, 5, 3, 1, 23, 0, 8, 0, 26, 12, 0, 0, 26

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A359270, A363035.

a[n_] := DivisorSum[5*n - 3, # + 2 &, Mod[#, 5] == 3 &]/5; Array[a, 100] (* Amiram Eldar, Jul 06 2023 *)

a(n) = sumdiv(5*n-3, d, (d%5==3)*(d+2))/5;

a(n) = (1/5) * Sum_{d | 5*n-3, d==3 (mod 5)} (d+2).

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

cp's OEIS Frontend

A359288 Number of divisors of 5*n-1 of form 5*k+3.

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A363035 Sum of divisors of 5*n-3 of form 5*k+3.

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A364102 Sum of divisors of 5*n-3 of form 5*k+4.

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A364106 Expansion of Sum_{k>0} k * x^(3*k) / (1 - x^(5*k-1)).

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A363157 Expansion of Sum_{k>0} k * x^(4*k-1) / (1 - x^(5*k-2)).

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A359288 Number of divisors of 5n-1 of form 5k+3.

A363035 Sum of divisors of 5n-3 of form 5k+3.

A364102 Sum of divisors of 5n-3 of form 5k+4.

A364106 Expansion of Sum_{k>0} k * x^(3k) / (1 - x^(5k-1)).

A363157 Expansion of Sum_{k>0} k * x^(4k-1) / (1 - x^(5k-2)).