A001878 -id:A001878 - 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)).

A363928 Expansion of Sum_{k>0} x^(3k) / (1 - x^(5k))^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 3, 0, 1, 2, 0, 5, 0, 0, 1, 0, 5, 3, 0, 3, 1, 6, 0, 1, 0, 2, 8, 0, 0, 5, 0, 8, 4, 2, 0, 1, 9, 0, 1, 5, 0, 13, 0, 0, 1, 3, 11, 5, 0, 8, 1, 12, 0, 1, 0, 0, 14, 2, 3, 8, 0, 14, 6, 0, 0, 7, 15, 0, 1, 8, 0, 20, 0, 2, 1, 0, 17, 7, 0, 9, 1, 20, 0, 5, 3, 5, 20, 0, 0, 13, 0, 20, 8, 0, 0

Offset: 1

Seiichi Manyama, Jun 28 2023

nonn

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

Cf. A363925, A363926, A363929.

Cf. A001878, A284281.

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

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

a(n) = (1/5) * Sum_{d|n, d==3 mod 5} (d+2) = (2 * A001878(n) + A284281(n))/5.

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

A364599 a(n) is the least number with exactly n divisors of the form 5*k+3.

Original entry on oeis.org

1, 3, 18, 48, 144, 288, 504, 1248, 1008, 2016, 3024, 4368, 5544, 14112, 12096, 17136, 11088, 34272, 22176, 26208, 33264, 52416, 48048, 91728, 66528, 117936, 155232, 183456, 133056, 235872, 188496, 222768, 144144, 528528, 376992, 616896, 421344, 825552, 288288, 1707552, 432432

Offset: 0

Ilya Gutkovskiy, Jul 29 2023

nonn

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

Cf. A001878, A005179, A038547, A364586, A364598, A364600.

list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = sumdiv(k, d, d % 5 == 3) + 1; if(i <= nmax+1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

A364045 Expansion of Sum_{k>0} x^(3k) / (1 + x^(5k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A364043, A364044, A364046.

Cf. A001878, A364021.

a[n_] := -DivisorSum[n, (-1)^# &, Mod[#, 5] == 3 &]; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)

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

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

a(n) = -Sum_{d|n, d==3 (mod 5)} (-1)^d.

A364419 Number of divisors of n of the form 5*k+3 that are at most sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 23 2023

nonn

Cf. A001878, A038548, A364388, A364389, A364420.

Table[Count[Divisors[n], _?(# <= Sqrt[n] && MemberQ[{3}, Mod[#, 5]] &)], {n, 99}]
nmax = 99; CoefficientList[Series[Sum[x^(5 k + 3)^2/(1 - x^(5 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest

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

cp's OEIS Frontend

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

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A363928 Expansion of Sum_{k>0} x^(3*k) / (1 - x^(5*k))^2.

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A364599 a(n) is the least number with exactly n divisors of the form 5*k+3.

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A364045 Expansion of Sum_{k>0} x^(3*k) / (1 + x^(5*k)).

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A364419 Number of divisors of n of the form 5*k+3 that are at most sqrt(n).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

A363928 Expansion of Sum_{k>0} x^(3k) / (1 - x^(5k))^2.

A364045 Expansion of Sum_{k>0} x^(3k) / (1 + x^(5k)).