A359237 -id:A359237 - cp's OEIS frontend

A359233 Number of divisors of 5n-1 of form 5k+1.

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

Offset: 1

Seiichi Manyama, Dec 22 2022

Also number of divisors of 5*n-1 of form 5*k+4.

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

Cf. A078703, A099774, A359211.

Cf. A001876, A001899, A359236, A359237, A359238.

a[n_] := DivisorSum[5*n-1, 1 &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

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

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

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(4*k-3)/(1-x^(5*k-4))))

a(n) = A001876(5*n-1) = A001899(5*n-1).
G.f.: Sum_{k>0} x^k/(1 - x^(5*k-1)).
G.f.: Sum_{k>0} x^(4*k-3)/(1 - x^(5*k-4)).

A359236 Number of divisors of 5n-2 of form 5k+1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 4, 1, 3, 2, 1, 1, 3, 1, 1, 2, 2, 2, 3, 1, 1, 3, 1, 1, 5, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 1, 3, 3, 1, 4, 1, 1, 2, 1, 2, 4, 1, 2, 2, 1, 1, 3, 1, 3

Offset: 1

Seiichi Manyama, Dec 22 2022

Also number of divisors of 5*n-2 of form 5*k+3.

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

Cf. A001876, A001878, A359233, A359237, A359238.

a[n_] := DivisorSum[5*n-2, 1 &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

PARI
```
a(n) = sumdiv(5*n-2, d, d%5==1);
```
PARI
```
a(n) = sumdiv(5*n-2, d, d%5==3);
```

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

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(3*k-2)/(1-x^(5*k-4))))

a(n) = A001876(5*n-2) = A001878(5*n-2).
G.f.: Sum_{k>0} x^k/(1 - x^(5*k-2)).
G.f.: Sum_{k>0} x^(3*k-2)/(1 - x^(5*k-4)).

A359238 Number of divisors of 5n-4 of form 5k+1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 3, 2, 2, 2, 4, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 2, 2, 2, 4, 2, 4, 2, 2, 3, 2

Offset: 1

Seiichi Manyama, Dec 22 2022

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

Cf. A001876, A359233, A359236, A359237.

a[n_] := DivisorSum[5*n-4, 1 &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

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

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(5*k-4))))

a(n) = A001876(5*n-4).

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

A359270 Number of divisors of 5n-3 of form 5k+3.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 4, 0, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 4, 0, 1, 1, 1, 0, 3, 0, 2, 1, 0, 0, 2, 1, 0, 2, 3, 0, 1, 0, 1, 1, 0, 0, 5, 1, 1, 1

Offset: 1

Seiichi Manyama, Dec 23 2022

Also number of divisors of 5*n-3 of form 5*k+4.

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

Cf. A001878, A001899, A359237.

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

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

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

my(N=100, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(4*k-1)/(1-x^(5*k-2)))))

a(n) = A001878(5*n-3) = A001899(5*n-3).
G.f.: Sum_{k>0} x^(3*k)/(1 - x^(5*k-1)).
G.f.: Sum_{k>0} x^(4*k-1)/(1 - x^(5*k-2)).

A359287 Number of divisors of 5n-1 of form 5k+2.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 24 2022

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

Cf. A001877.
Cf. A359233, A359288.
Cf. A359237, A359244, A359269.

a[n_] := DivisorSum[5*n-1, 1 &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(5*n-1, d, d%5==2);
```

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

a(n) = A001877(5*n-1).

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

A363026 Sum of divisors of 5n-3 of form 5k+2.

Original entry on oeis.org

2, 7, 14, 17, 24, 27, 34, 37, 51, 47, 54, 57, 64, 67, 86, 84, 84, 87, 94, 97, 121, 107, 121, 117, 124, 127, 168, 137, 144, 154, 154, 157, 191, 167, 174, 177, 191, 204, 238, 197, 204, 207, 214, 224, 261, 227, 234, 237, 266, 247, 315, 257, 264, 267, 291, 277, 331, 294, 294, 324, 304, 307, 378, 317, 331, 327

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A362952, A363025, A363027.

Cf. A284280, A359237, A363030.

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

PARI
```
a(n) = sumdiv(5*n-3, d, (d%5==2)*d);
```

a(n) = A284280(5*n-3).

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

A364094 Sum of divisors of 5n-3 of form 5k+1.

Original entry on oeis.org

1, 1, 7, 1, 12, 1, 17, 1, 28, 1, 27, 1, 32, 1, 43, 12, 42, 1, 47, 1, 58, 1, 73, 1, 62, 1, 84, 1, 72, 22, 77, 1, 88, 1, 87, 1, 118, 12, 119, 1, 102, 1, 107, 32, 118, 1, 117, 1, 133, 1, 190, 1, 132, 1, 153, 1, 148, 42, 147, 12, 152, 1, 189, 1, 208, 1, 167, 1, 178, 1, 204, 73, 182, 1, 224, 1, 192, 1

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364092, A364093, A364095.

Cf. A284097, A359237, A364098.

a[n_] := DivisorSum[5*n - 3, # &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

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

a(n) = A284097(5*n-3).

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

A363030 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k-3)).

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 8, 8, 12, 10, 12, 12, 14, 14, 19, 18, 18, 18, 20, 20, 26, 22, 26, 24, 26, 26, 36, 28, 30, 32, 32, 32, 40, 34, 36, 36, 40, 42, 50, 40, 42, 42, 44, 46, 54, 46, 48, 48, 55, 50, 66, 52, 54, 54, 60, 56, 68, 60, 60, 66, 62, 62, 78, 64, 68, 66, 68, 68, 82, 70, 84, 78, 74, 74, 92, 76, 78

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A359237, A363026.

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

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

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

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

A364098 Expansion of Sum_{k>0} k * x^(2k-1) / (1 - x^(5k-4)).

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 5, 1, 8, 1, 7, 1, 8, 1, 11, 4, 10, 1, 11, 1, 14, 1, 17, 1, 14, 1, 20, 1, 16, 6, 17, 1, 20, 1, 19, 1, 26, 4, 27, 1, 22, 1, 23, 8, 26, 1, 25, 1, 29, 1, 42, 1, 28, 1, 33, 1, 32, 10, 31, 4, 32, 1, 41, 1, 44, 1, 35, 1, 38, 1, 44, 17, 38, 1, 48, 1, 40, 1, 53, 1, 44, 4, 43, 1, 44, 14, 59, 1

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A359237, A364094.

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

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

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

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

cp's OEIS Frontend

A359233 Number of divisors of 5*n-1 of form 5*k+1.

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A359236 Number of divisors of 5*n-2 of form 5*k+1.

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A359238 Number of divisors of 5*n-4 of form 5*k+1.

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A359270 Number of divisors of 5*n-3 of form 5*k+3.

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A359287 Number of divisors of 5*n-1 of form 5*k+2.

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A363026 Sum of divisors of 5*n-3 of form 5*k+2.

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A364094 Sum of divisors of 5*n-3 of form 5*k+1.

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A359233 Number of divisors of 5n-1 of form 5k+1.

A359236 Number of divisors of 5n-2 of form 5k+1.

A359238 Number of divisors of 5n-4 of form 5k+1.

A359270 Number of divisors of 5n-3 of form 5k+3.

A359287 Number of divisors of 5n-1 of form 5k+2.

A363026 Sum of divisors of 5n-3 of form 5k+2.

A364094 Sum of divisors of 5n-3 of form 5k+1.

A364098 Expansion of Sum_{k>0} k * x^(2k-1) / (1 - x^(5k-4)).