A001877 -id:A001877 - cp's OEIS frontend

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

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)).

A363926 Expansion of Sum_{k>0} x^(2k) / (1 - x^(5k))^2.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 4, 0, 3, 0, 1, 4, 1, 0, 1, 2, 6, 0, 4, 0, 1, 6, 3, 0, 1, 0, 8, 0, 5, 2, 4, 8, 1, 0, 1, 0, 12, 0, 6, 0, 1, 10, 4, 2, 1, 4, 12, 0, 7, 0, 3, 12, 1, 0, 4, 0, 14, 2, 8, 0, 6, 14, 5, 0, 3, 0, 19, 0, 9, 0, 1, 18, 1, 0, 1, 6, 18, 0, 15, 4, 1, 18, 6, 0, 1, 2, 20, 0

Offset: 1

Seiichi Manyama, Jun 28 2023

nonn

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

Cf. A363925, A363928, A363929.

Cf. A001877, A284280.

a[n_] := DivisorSum[n, # + 3 &, Mod[#, 5] == 2 &] / 5; Array[a, 100] (* Amiram Eldar, Jun 28 2023 *)

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

a(n) = (1/5) * Sum_{d|n, d==2 mod 5} (d+3) = (3 * A001877(n) + A284280(n))/5.

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

A364598 a(n) is the least number with exactly n divisors of the form 5*k+2.

Original entry on oeis.org

1, 2, 12, 42, 84, 252, 462, 672, 924, 2016, 2772, 4032, 5544, 9072, 7392, 17136, 14784, 26208, 22176, 34272, 33264, 52416, 44352, 119952, 66528, 117936, 99792, 183456, 125664, 222768, 188496, 235872, 199584, 487872, 288288, 616896, 399168, 1206576, 376992, 1097712, 432432

Offset: 0

Ilya Gutkovskiy, Jul 29 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 0..100

Cf. A001877, A005179, A038547, A364586, A364599, A364600.

list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = sumdiv(k, d, d % 5 == 2) + 1; if(i <= nmax+1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

A364044 Expansion of Sum_{k>0} x^(2k) / (1 + x^(5k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A364043, A364045, A364046.

Cf. A001877, A364020.

a[n_] := DivisorSum[n, (-1)^# &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

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

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

a(n) = Sum_{d|n, d==2 (mod 5)} (-1)^d.

A364389 Number of divisors of n of the form 5*k+2 that are at most sqrt(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1

Offset: 1

Ilya Gutkovskiy, Jul 21 2023

nonn

Cf. A001877, A038548, A364388.

Table[Count[Divisors[n], _?(# <= Sqrt[n] && MemberQ[{2}, Mod[#, 5]] &)], {n, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(5 k + 2)^2/(1 - x^(5 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest

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

A373335 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4*k)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 2, 0, 1, -1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, -1, 1, 1, 2, 0, 2, -1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 2, 0, -1, 1, -1, 2, 0, 1, -1, 1, 1, 0, 2, 1, 1, 1, 0, 1, -1, 0, 1, 0, 0, 1, 1, 1, 0, 2, -1, 1, 1, 0, -1, 2, 0, 2

Offset: 1

Seiichi Manyama, Jun 01 2024

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

Cf. A002324, A048272, A373336.

Cf. A001876, A001877.

my(N=110, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1-x^(5*k))))

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

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

a(n) = A001876(n) - A001877(n).

cp's OEIS Frontend

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

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A363926 Expansion of Sum_{k>0} x^(2*k) / (1 - x^(5*k))^2.

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A364598 a(n) is the least number with exactly n divisors of the form 5*k+2.

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A364044 Expansion of Sum_{k>0} x^(2*k) / (1 + x^(5*k)).

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A364389 Number of divisors of n of the form 5*k+2 that are at most sqrt(n).

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Author

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Crossrefs

Programs

Mathematica

Formula

A373335 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k)).

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Author

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PARI

PARI

Formula

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

A363926 Expansion of Sum_{k>0} x^(2k) / (1 - x^(5k))^2.

A364044 Expansion of Sum_{k>0} x^(2k) / (1 + x^(5k)).

A373335 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4*k)).