A001876 -id:A001876 - cp's OEIS frontend

A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

1, 2, 3, 4, 5, 7, 7, 8, 9, 10, 12, 14, 13, 14, 15, 17, 17, 21, 19, 20, 22, 24, 23, 28, 25, 27, 27, 28, 29, 35, 32, 34, 36, 34, 35, 43, 37, 38, 39, 40, 42, 51, 43, 48, 45, 47, 47, 59, 49, 50, 52, 54, 53, 63, 60, 57, 57, 58, 59, 70, 62, 64, 66, 68, 65, 84, 67, 68, 69, 70, 72, 86, 73, 74, 75, 77, 84, 94

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A363898, A363899, A363900.
Cf. A002131, A050460, A326399.
Cf. A001876, A284097.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

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

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

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

A363925 Expansion of Sum_{k>0} x^k / (1 - x^(5*k))^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 5, 1, 3, 1, 1, 6, 4, 1, 3, 1, 7, 1, 1, 1, 3, 8, 5, 4, 1, 1, 11, 1, 1, 1, 1, 10, 8, 1, 4, 1, 11, 1, 7, 1, 1, 12, 7, 1, 3, 4, 13, 1, 1, 1, 3, 14, 8, 6, 5, 1, 20, 1, 1, 1, 1, 16, 11, 1, 1, 1, 17, 4, 9, 1, 5, 18, 10, 1, 8, 1, 19, 1, 4, 1, 3, 20, 11

Offset: 1

Seiichi Manyama, Jun 28 2023

nonn

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

Cf. A363926, A363928, A363929.
Cf. A001876, A284097.
Cf. A363901, A363903.

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

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

a(n) = (1/5) * Sum_{d|n, d==1 mod 5} (d+4) = (4 * A001876(n) + A284097(n))/5.

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

A218444 a(n) = Sum_{k>=0} floor(n/(5*k + 1)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 20, 21, 23, 24, 25, 27, 29, 30, 32, 33, 35, 36, 37, 38, 40, 42, 44, 46, 47, 48, 51, 52, 53, 54, 55, 57, 60, 61, 63, 64, 66, 67, 70, 71, 72, 74, 76, 77, 79, 81, 83, 84, 85, 86, 88, 90, 92, 94, 96, 97, 101, 102, 103, 104, 105

Offset: 0

Benoit Cloitre, Oct 28 2012

Seiichi Manyama, 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 A001876.
Cf. A218444, A218445, A218446.
Cf. A001620, A256779.

a[n_] := Sum[ Floor[n/(5*k+1)], {k, 0, Ceiling[n/5]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 22 2013 *)

A218444[n]:=sum(floor(n/(5*k+1)),k,0,n)$
makelist(A218444[n],n,0,80); /* Martin Ettl, Oct 29 2012 */

PARI
```
a(n)=sum(k=0,n,(n\(5*k+1)))
```

a(n) = Sum_{k>=0} floor(n/(5*k + 1)).

a(n) = 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, Apr 20 2025

A364043 Expansion of Sum_{k>0} x^k / (1 + x^(5*k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A364044, A364045, A364046.
Cf. A002654, A363037, A364011, A364047.
Cf. A001876, A364019.

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

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

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

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

A364388 Number of divisors of n of the form 5*k+1 that are at most sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Ilya Gutkovskiy, Jul 21 2023

nonn

Cf. A001876, A038548, A364389.

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

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

A364586 a(n) is the least number with exactly n divisors of the form 5*k+1.

Original entry on oeis.org

1, 6, 36, 66, 252, 336, 672, 1008, 3528, 2016, 4032, 3696, 9072, 7392, 13104, 11088, 36288, 38304, 26208, 22176, 68544, 44352, 91728, 66528, 154224, 99792, 209664, 96096, 301392, 144144, 222768, 188496, 487872, 399168, 471744, 421344, 1079568, 288288, 1097712, 432432

Offset: 1

Ilya Gutkovskiy, Jul 28 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A001876, A005179, A038547, A364598, A364599, A364600.

a(n) = my(k=1); while (sumdiv(k, d, (d%5)==1) != n, k++); k; \\ Michel Marcus, Jul 29 2023

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

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

A363897 Expansion of Sum_{k>0} k * x^k / (1 - x^(5*k)).

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A363925 Expansion of Sum_{k>0} x^k / (1 - x^(5*k))^2.

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A218444 a(n) = Sum_{k>=0} floor(n/(5*k + 1)).

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Maxima

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A364043 Expansion of Sum_{k>0} x^k / (1 + x^(5*k)).

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A364388 Number of divisors of n of the form 5*k+1 that are at most sqrt(n).

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A364586 a(n) is the least number with exactly n divisors of the form 5*k+1.

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

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