A284097 -id:A284097 - cp's OEIS frontend

A284150 Sum_{d|n, d==1 or 4 mod 5} d.

1, 1, 1, 5, 1, 7, 1, 5, 10, 1, 12, 11, 1, 15, 1, 21, 1, 16, 20, 5, 22, 12, 1, 35, 1, 27, 10, 19, 30, 7, 32, 21, 12, 35, 1, 56, 1, 20, 40, 5, 42, 42, 1, 60, 10, 47, 1, 51, 50, 1, 52, 31, 1, 70, 12, 75, 20, 30, 60, 11, 62, 32, 31, 85, 1, 84, 1, 39, 70, 15, 72, 80, 1

Offset: 1

Seiichi Manyama, Mar 21 2017

nonn

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

Cf. A003114, A284097, A284103.

Cf. Sum_{d|n, d==1 or k-1 mod k} d: A046913 (k=3), A000593 (k=4), this sequence (k=5), A186099 (k=6), A284151 (k=7).

A284150 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,5) in {1,4} then
            a := a+d ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 21 2017

Table[Sum[If[Mod[d, 5] == 1 || Mod[d,5]==4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 80, print1(sumdiv(n, d, if(d%5==1 || d%5 ==4, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%5==1 or d%5 == 4]) # Indranil Ghosh, Mar 21 2017

a(n) = A284097(n) + A284103(n). - Seiichi Manyama, Mar 24 2017

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

Original entry on oeis.org

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)).

A364092 Sum of divisors of 5n-1 of form 5k+1.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 1, 12, 1, 7, 1, 17, 1, 1, 1, 28, 1, 1, 12, 27, 1, 7, 1, 32, 1, 1, 1, 59, 1, 12, 1, 42, 1, 7, 1, 47, 22, 1, 1, 58, 12, 1, 1, 73, 1, 33, 1, 62, 1, 1, 1, 84, 1, 1, 32, 72, 1, 28, 1, 93, 1, 1, 12, 124, 1, 1, 1, 87, 1, 7, 1, 118, 42, 12, 1, 119, 1, 1, 22, 102, 1, 53, 1, 107, 12, 32, 1

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364093, A364094, A364095.
Cf. A008438, A363514.
Cf. A284097, A359233, A364096.

a[n_] := DivisorSum[5*n - 1, # &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

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

a(n) = A284097(5*n-1).

G.f.: Sum_{k>0} (5*k-4) * x^(4*k-3) / (1 - x^(5*k-4)).

A364093 Sum of divisors of 5n-2 of form 5k+1.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 12, 1, 1, 23, 1, 1, 22, 1, 1, 33, 1, 12, 32, 1, 1, 43, 1, 1, 42, 17, 1, 53, 12, 1, 52, 1, 1, 84, 1, 1, 62, 1, 1, 84, 1, 43, 72, 1, 1, 83, 1, 1, 82, 32, 12, 93, 1, 1, 113, 1, 1, 155, 1, 1, 102, 12, 1, 113, 1, 42, 112, 27, 1, 123, 1, 1, 133, 63, 1, 154, 1, 1, 132, 1, 32, 194, 1, 12, 142

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364092, A364094, A364095.

Cf. A284097, A359236, A364097.

a[n_] := DivisorSum[5*n - 2, # &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

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

a(n) = A284097(5*n-2).

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

A364094 Sum of divisors of 5n-3 of form 5k+1.

Original entry on oeis.org

1, 1, 7, 1, 12, 1, 17, 1, 28, 1, 27, 1, 32, 1, 43, 12, 42, 1, 47, 1, 58, 1, 73, 1, 62, 1, 84, 1, 72, 22, 77, 1, 88, 1, 87, 1, 118, 12, 119, 1, 102, 1, 107, 32, 118, 1, 117, 1, 133, 1, 190, 1, 132, 1, 153, 1, 148, 42, 147, 12, 152, 1, 189, 1, 208, 1, 167, 1, 178, 1, 204, 73, 182, 1, 224, 1, 192, 1

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364092, A364093, A364095.

Cf. A284097, A359237, A364098.

a[n_] := DivisorSum[5*n - 3, # &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)

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

a(n) = A284097(5*n-3).

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

A364095 Sum of divisors of 5n-4 of form 5k+1.

Original entry on oeis.org

1, 7, 12, 17, 22, 27, 32, 43, 42, 47, 52, 57, 62, 84, 72, 77, 82, 87, 92, 119, 102, 107, 112, 117, 133, 154, 132, 137, 142, 147, 152, 189, 162, 167, 172, 204, 182, 224, 192, 197, 202, 207, 212, 259, 222, 227, 264, 237, 242, 294, 252, 273, 262, 267, 272, 329, 282, 324, 292, 297, 302, 364, 312

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Cf. A364092, A364093, A364094.

Cf. A284097, A359238, A364099.

a[n_] := DivisorSum[5*n - 4, # &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 12 2023 *)

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

a(n) = A284097(5*n-4).

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

A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 24, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 46, 24, 1, 13, 1, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 68, 35, 24, 1, 1, 13, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 13, 1

Offset: 1

Seiichi Manyama, Jan 17 2023

nonn

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

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8), this sequence (k=11).

Cf. A357911.

a[n_] := DivisorSum[n, # &, Mod[#, 11] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 09 2023 *)

a(n) = sumdiv(n, d, (Mod(d, 11)==1)*d);

my(N=100, x='x+O('x^N)); Vec(sum(k=0, N, (11*k+1)*x^(11*k+1)/(1-x^(11*k+1))))

G.f.: Sum_{k>=0} (11*k+1) * x^(11*k+1)/(1 - x^(11*k+1)).

cp's OEIS Frontend

A284150 Sum_{d|n, d==1 or 4 mod 5} d.

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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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A364092 Sum of divisors of 5*n-1 of form 5*k+1.

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A364093 Sum of divisors of 5*n-2 of form 5*k+1.

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A364094 Sum of divisors of 5*n-3 of form 5*k+1.

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A364095 Sum of divisors of 5*n-4 of form 5*k+1.

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Mathematica

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A357912 a(n) = Sum_{d|n, d==1 (mod 11)} d.

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A364092 Sum of divisors of 5n-1 of form 5k+1.

A364093 Sum of divisors of 5n-2 of form 5k+1.

A364094 Sum of divisors of 5n-3 of form 5k+1.

A364095 Sum of divisors of 5n-4 of form 5k+1.