A001899 -id:A001899 - cp's OEIS frontend

A218447 a(n) = Sum_{k>=0} floor(n/(5*k + 4)).

0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 9, 9, 11, 11, 11, 12, 14, 15, 15, 15, 16, 16, 17, 17, 19, 19, 20, 21, 22, 22, 23, 23, 25, 26, 26, 26, 28, 29, 29, 29, 30, 30, 32, 32, 34, 35, 36, 37, 38, 38, 38, 39, 41, 41, 41, 41, 43, 44, 45, 45, 48, 48, 49, 49, 51, 51, 52, 53, 54

Offset: 0

Benoit Cloitre, Oct 28 2012

Robert Israel, 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.

Partial sums of A001899.
Cf. A218444, A218445, A218446.
Cf. A001620, A256849.

g:= n -> nops(select(t -> t mod 5 = 4, numtheory:-divisors(n))):
g(0):= 0:
ListTools:-PartialSums(map(g, [$0..100])); # Robert Israel, Apr 29 2021

A218447[n]:=sum(floor(n/(5*k+4)),k,0,n)$
makelist(A218447[n],n,0,80); /* Martin Ettl, Oct 20 2012 */

PARI
```
a(n)=sum(k=0,n,(n\(5*k+4)))
```

a(n) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,5) - (1 - gamma)/5 = A256849 - (1 - A001620)/5 = -0.213442... (Smith and Subbarao, 1981). - Amiram Eldar, Apr 20 2025

A364046 Expansion of Sum_{k>0} x^(4k) / (1 + x^(5k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A364043, A364044, A364045.

Cf. A001899, A364022.

a[n_] := DivisorSum[n, (-1)^# &, Mod[#, 5] == 4 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

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

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

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

A364420 Number of divisors of n of the form 5*k+4 that are at most sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 23 2023

nonn

Cf. A001899, A038548, A364388, A364389, A364419.

Table[Count[Divisors[n], _?(# <= Sqrt[n] && MemberQ[{4}, Mod[#, 5]] &)], {n, 108}]
nmax = 108; CoefficientList[Series[Sum[x^(5 k + 4)^2/(1 - x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x] // Rest

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

cp's OEIS Frontend

A218447 a(n) = Sum_{k>=0} floor(n/(5*k + 4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Maxima

PARI

Formula

A364046 Expansion of Sum_{k>0} x^(4k) / (1 + x^(5k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A364420 Number of divisors of n of the form 5*k+4 that are at most sqrt(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A218447 a(n) = Sum_{k>=0} floor(n/(5*k + 4)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Maxima

PARI

Formula

A364046 Expansion of Sum_{k>0} x^(4*k) / (1 + x^(5*k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A364420 Number of divisors of n of the form 5*k+4 that are at most sqrt(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A364046 Expansion of Sum_{k>0} x^(4k) / (1 + x^(5k)).