A001899 -id:A001899 - cp's OEIS frontend

A001876 Number of divisors of n of the form 5k+1; a(0)=0.

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

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

For numbers of divisors of n of the form 5k+i (i=1, 2, 3, 4) see: this sequence, A001877, A001878, A001899.

Cf. A001620, A256779.

CoefficientList[ Series[ Together[ Sum[ x^n/(1 - x^(5n)), {n, 110}]], {x, 0, 110}], x] (* Robert G. Wilson v, Jan 31 2011 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 5] == 1 &]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

a(n) = if(n==0,0, sumdiv(n, d, (d % 5) == 1)); \\ Michel Marcus, Feb 25 2021

G.f.: Sum_{n>=0} x^(5n+1)/(1-x^(5n+1)).
G.f.: Sum_{n>=1} x^n/(1-x^(5*n)). - Joerg Arndt, Jan 30 2011
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,5) - (1 - gamma)/5 = A256779 - (1 - A001620)/5 = 0.651363... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A001877 Number of divisors of n of the form 5k+2; a(0) = 0.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001876, A001878, A001899.

Cf. A001620, A256780.

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 2 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
Table[Count[Divisors[n],?(Mod[#,5]==2&)],{n,0,90}] (* _Harvey P. Dale, May 20 2017 *)

a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==2)); \\ Michel Marcus, Feb 28 2021

G.f.: Sum_{n>=0} x^(5n+2)/(1-x^(5n+2)).
G.f.: Sum_{n>=1} x^(2*n)/(1-x^(5*n)). - Joerg Arndt, Jan 30 2011
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,5) - (1 - gamma)/5 = A256780 - (1 - A001620)/5 = 0.105832... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A001878 Number of divisors of n of the form 5k+3; a(0) = 0.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..10000
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001876, A001877, A001899.

Cf. A001620, A256848.

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 3 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)
Table[Count[Divisors[n],?(Mod[#,5]==3&)],{n,0,90}] (* _Harvey P. Dale, Nov 08 2012 *)

a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==3)); \\ Michel Marcus, Feb 28 2021

G.f.: Sum_{n>=0} x^(5*n+3)/(1 - x^(5*n+3)).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,5) - (1 - gamma)/5 = A256848 - (1 - A001620)/5 = -0.0983206... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

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

Original entry on oeis.org

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)).

A359241 Number of divisors of 5n-4 of form 5k+4.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 22 2022

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

Cf. A359239, A359240.

Cf. A001899.

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

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

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

a(n) = A001899(5*n-4).

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

A359269 Number of divisors of 5n-2 of form 5k+2.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 23 2022

Also number of divisors of 5*n-2 of form 5*k+4.

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

Cf. A001877, A001899, A359236.

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

PARI
```
a(n) = sumdiv(5*n-2, d, d%5==2);
```
PARI
```
a(n) = sumdiv(5*n-2, d, d%5==4);
```

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

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(4*k-2)/(1-x^(5*k-3)))))

a(n) = A001877(5*n-2) = A001899(5*n-2).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(5*k-1)).
G.f.: Sum_{k>0} x^(4*k-2)/(1 - x^(5*k-3)).

A363900 Expansion of Sum_{k>0} k * x^(4k) / (1 - x^(5k)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 3, 0, 1, 0, 4, 0, 2, 1, 5, 0, 0, 0, 7, 0, 0, 3, 9, 1, 0, 0, 8, 0, 1, 0, 13, 0, 2, 1, 10, 0, 3, 0, 12, 5, 0, 0, 14, 1, 0, 0, 13, 0, 7, 0, 18, 3, 2, 1, 15, 0, 0, 7, 17, 0, 0, 0, 19, 1, 5, 0, 29, 0, 1, 0, 23, 0, 2, 1, 20, 9, 0, 0, 28, 0, 0, 3, 24, 1, 10, 0, 23, 0, 1, 5, 28, 0

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A363897, A363898, A363899.

Cf. A001899, A284103.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

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

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

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

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)).

A363929 Expansion of Sum_{k>0} x^(4k) / (1 - x^(5k))^2.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 3, 0, 1, 0, 2, 4, 1, 0, 0, 0, 6, 0, 0, 2, 4, 6, 0, 0, 1, 0, 7, 0, 3, 0, 4, 8, 1, 0, 3, 0, 10, 2, 0, 0, 6, 10, 0, 0, 1, 0, 13, 0, 4, 4, 6, 12, 1, 0, 0, 2, 14, 0, 0, 0, 8, 14, 3, 0, 8, 0, 15, 0, 5, 0, 8, 16, 1, 2, 0, 0, 21, 0, 0, 6, 10, 18, 2, 0, 1, 0, 19, 4, 6, 0, 13, 22, 1, 0, 7, 0, 22, 0, 0, 0, 14, 22, 0, 0

Offset: 1

Seiichi Manyama, Jun 28 2023

nonn

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

Cf. A363925, A363926, A363928.

Cf. A001899, A284103.

a[n_] := DivisorSum[n, # + 1 &, Mod[#, 5] == 4 &] / 5; Array[a, 100] (* Amiram Eldar, Jun 28 2023 *)

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

a(n) = (1/5) * Sum_{d|n, d==4 mod 5} (d+1) = (A001899(n) + A284103(n))/5.

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

A364600 a(n) is the least number with exactly n divisors of the form 5*k+4.

Original entry on oeis.org

1, 4, 24, 72, 144, 432, 504, 1008, 1512, 3528, 3024, 7056, 5544, 14112, 11088, 13104, 16632, 26208, 36288, 63504, 33264, 68544, 77616, 127008, 66528, 154224, 155232, 209664, 133056, 222768, 144144, 301392, 216216, 853776, 288288, 471744, 399168, 825552, 698544, 1707552, 432432

Offset: 0

Ilya Gutkovskiy, Jul 29 2023

nonn

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

Cf. A001899, A005179, A038547, A364586, A364598, A364599.

list(nmax) = {my(v = vector(nmax+1), c = 0, k = 1, i); while(c < nmax+1, i = sumdiv(k, d, d % 5 == 4) + 1; if(i <= nmax+1 && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Jan 28 2025

cp's OEIS Frontend

A001876 Number of divisors of n of the form 5k+1; a(0)=0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A001877 Number of divisors of n of the form 5k+2; a(0) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A001878 Number of divisors of n of the form 5k+3; a(0) = 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A359241 Number of divisors of 5*n-4 of form 5*k+4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A359269 Number of divisors of 5*n-2 of form 5*k+2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A363900 Expansion of Sum_{k>0} k * x^(4*k) / (1 - x^(5*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

A359241 Number of divisors of 5n-4 of form 5k+4.

A359269 Number of divisors of 5n-2 of form 5k+2.

A363900 Expansion of Sum_{k>0} k * x^(4k) / (1 - x^(5k)).

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

A363929 Expansion of Sum_{k>0} x^(4k) / (1 - x^(5k))^2.