A359233 -id:A359233 - cp's OEIS frontend

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

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)).

A359237 Number of divisors of 5n-3 of form 5k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 22 2022

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

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

Cf. A001876, A001877, A359233, A359236, A359238.

a[n_] := DivisorSum[5*n-3, 1 &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 23 2023 *)

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

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

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

a(n) = A001876(5*n-3) = A001877(5*n-3).
G.f.: Sum_{k>0} x^k/(1 - x^(5*k-3)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(5*k-4)).

A359305 Number of divisors of 6n-1 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Also number of divisors of 6*n-1 of form 6*k+5.

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

Cf. A279060, A319995.
Cf. A000005, A001620, A016969, A078703, A359211, A359233.
Cf. A359306, A359307, A359308, A359309.
Cf. A359324, A359325, A359326, A359327.

a[n_] := DivisorSigma[0, 6*n - 1]/2; Array[a, 100] (* Amiram Eldar, Dec 26 2022 *)

PARI
```
a(n) = sumdiv(6*n-1, d, d%6==1);
```
PARI
```
a(n) = sumdiv(6*n-1, d, d%6==5);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-1))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(5*k-4)/(1-x^(6*k-5))))

a(n) = A279060(6*n-1) = A319995(6*n-1).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-1)).
G.f.: Sum_{k>0} x^(5*k-4)/(1 - x^(6*k-5)).
From Amiram Eldar, Dec 26 2022: (Start)
a(n) = A000005(A016969(n-1))/2.
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 3*log(2) + 2*log(3))*n/6 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). (End)

A364104 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k-1)).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 10, 10, 13, 12, 14, 14, 15, 16, 21, 18, 19, 22, 22, 22, 27, 24, 26, 26, 27, 28, 37, 30, 34, 32, 34, 34, 41, 36, 38, 40, 39, 40, 49, 46, 43, 44, 49, 46, 57, 48, 50, 50, 51, 52, 68, 54, 55, 58, 58, 58, 72, 60, 66, 62, 63, 70, 79, 66, 67, 68, 70, 70, 83, 72, 77, 76, 82, 76, 96

Offset: 1

Seiichi Manyama, Jul 05 2023

nonn

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

Cf. A364105, A364106, A364107.
Cf. A363028, A363155, A364096.
Cf. A359233, A364100.

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

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

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

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

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)).

A364096 Expansion of Sum_{k>0} k * x^(4k-3) / (1 - x^(5k-4)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 5, 1, 1, 1, 8, 1, 1, 4, 7, 1, 3, 1, 8, 1, 1, 1, 15, 1, 4, 1, 10, 1, 3, 1, 11, 6, 1, 1, 14, 4, 1, 1, 17, 1, 9, 1, 14, 1, 1, 1, 20, 1, 1, 8, 16, 1, 8, 1, 21, 1, 1, 4, 28, 1, 1, 1, 19, 1, 3, 1, 26, 10, 4, 1, 27, 1, 1, 6, 22, 1, 13, 1, 23, 4, 8, 1, 26, 1, 1, 12, 29, 1, 3, 1

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

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

Cf. A359233, A364092.

Cf. A363028, A363155, A364104.

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

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

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

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

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

Original entry on oeis.org

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)).

A364092 Sum of divisors of 5n-1 of form 5k+1.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 1, 12, 1, 7, 1, 17, 1, 1, 1, 28, 1, 1, 12, 27, 1, 7, 1, 32, 1, 1, 1, 59, 1, 12, 1, 42, 1, 7, 1, 47, 22, 1, 1, 58, 12, 1, 1, 73, 1, 33, 1, 62, 1, 1, 1, 84, 1, 1, 32, 72, 1, 28, 1, 93, 1, 1, 12, 124, 1, 1, 1, 87, 1, 7, 1, 118, 42, 12, 1, 119, 1, 1, 22, 102, 1, 53, 1, 107, 12, 32, 1

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364093, A364094, A364095.
Cf. A008438, A363514.
Cf. A284097, A359233, A364096.

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

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

a(n) = A284097(5*n-1).

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

A364100 Sum of divisors of 5n-1 of form 5k+4.

Original entry on oeis.org

4, 9, 14, 19, 28, 29, 34, 39, 48, 49, 63, 59, 68, 69, 74, 79, 102, 89, 94, 108, 108, 109, 133, 119, 128, 129, 134, 139, 181, 149, 168, 159, 168, 169, 203, 179, 188, 198, 194, 199, 242, 228, 214, 219, 242, 229, 282, 239, 248, 249, 254, 259, 336, 269, 274, 288, 288, 289, 357, 299, 327, 309, 314, 348

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364101, A364102, A364103.

Cf. A284103, A359233, A364104.

a[n_] := DivisorSum[5*n - 1, # &, Mod[#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

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

a(n) = A284103(5*n-1).

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

cp's OEIS Frontend

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

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A359237 Number of divisors of 5*n-3 of form 5*k+1.

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A359305 Number of divisors of 6*n-1 of form 6*k+1.

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A364104 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k-1)).

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

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A364096 Expansion of Sum_{k>0} k * x^(4*k-3) / (1 - x^(5*k-4)).

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

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

A359237 Number of divisors of 5n-3 of form 5k+1.

A359305 Number of divisors of 6n-1 of form 6k+1.

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

A364096 Expansion of Sum_{k>0} k * x^(4k-3) / (1 - x^(5k-4)).

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

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

A364092 Sum of divisors of 5n-1 of form 5k+1.

A364100 Sum of divisors of 5n-1 of form 5k+4.