A284280 -id:A284280 - cp's OEIS frontend

A109091 Expansion of (1 - eta(q)^5 / eta(q^5)) / 5 in powers of q.

1, -1, -2, 3, 1, 2, -6, -5, 7, -1, 12, -6, -12, 6, -2, 11, -16, -7, 20, 3, 12, -12, -22, 10, 1, 12, -20, -18, 30, 2, 32, -21, -24, 16, -6, 21, -36, -20, 24, -5, 42, -12, -42, 36, 7, 22, -46, -22, 43, -1, 32, -36, -52, 20, 12, 30, -40, -30, 60, -6, 62, -32, -42, 43, -12, 24, -66, -48, 44, 6, 72, -35, -72, 36, -2, 60, -72

Offset: 1

Michael Somos, Jun 18 2005

			G.f. = q - q^2 - 2*q^3 + 3*q^4 + q^5 + 2*q^6 - 6*q^7 - 5*q^8 + 7*q^9 - q^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This is the expression B^5/C in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]

Cf. A003823, A007325, A078905, A109064, A229793.

Cf. A284097, A284103, A284150, A284152, A284280, A284281.

a[ n_] := If[ n < 1, 0, Sum[ d KroneckerSymbol[ 5, d], {d, Divisors@n}]]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^5 / QPochhammer[ q^5]) / 5, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

{a(n) = my(A); if( n<1, 0, A = x * O(x^n); -1/5 * polcoeff( eta(x + A)^5 / eta(x^5 + A), n))};

{a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(5, d)))} /* Michael Somos, Mar 21 2008 */

def s(k, m, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == m}
  s
end
def A109091(n)
  (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
end # Seiichi Manyama, Apr 01 2017

G.f.: (1 - Product_{k>0} (1 - x^k)^5 / (1 - x^(5*k))) / 5 = Sum_{k>0} x^k * (1 - x^k)^2 * (1 + x^(6*k) - 4*x^(2*k) * (1 + x^k +x^(2*k))) / (1 - x^(5*k))^2.
-5*a(n) = A109064(n) unless n = 0.
a(n) = A284097(n) + A284103(n) - A284280(n) - A284281(n) = A284150(n) - A284152(n). - Seiichi Manyama, Apr 01 2017
L.g.f.: -log(1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(15*sqrt(5)) = 0.294254... . - Amiram Eldar, Jan 29 2024

A284281 a(n) = Sum_{d|n, d == 3 (mod 5)} d.

Original entry on oeis.org

0, 0, 3, 0, 0, 3, 0, 8, 3, 0, 0, 3, 13, 0, 3, 8, 0, 21, 0, 0, 3, 0, 23, 11, 0, 13, 3, 28, 0, 3, 0, 8, 36, 0, 0, 21, 0, 38, 16, 8, 0, 3, 43, 0, 3, 23, 0, 59, 0, 0, 3, 13, 53, 21, 0, 36, 3, 58, 0, 3, 0, 0, 66, 8, 13, 36, 0, 68, 26, 0, 0, 29, 73, 0, 3, 38, 0, 94, 0, 8

Offset: 1

Seiichi Manyama, Mar 24 2017

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

Cf. A013661, A109699, A284152.

Cf. Sum_{d|n, d=k mod 5} d: A284097 (k=1), A284280 (k=2), this sequence (k=3), A284103 (k=4).

Table[Sum[If[Mod[d, 5] == 3, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 24 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 5)==3, d, 0)),", ")) \\ Indranil Ghosh, Mar 24 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%5==3]) # Indranil Ghosh, Mar 24 2017

G.f.: Sum_{k>=0} (5*k + 3)*x^(5*k+3)/(1 - x^(5*k+3)). - Ilya Gutkovskiy, Mar 25 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/60 = 0.164493... (A013661 / 10). - Amiram Eldar, Nov 26 2023

A284152 a(n) = Sum_{d|n, d == 2 or 3 mod 5} d.

Original entry on oeis.org

0, 2, 3, 2, 0, 5, 7, 10, 3, 2, 0, 17, 13, 9, 3, 10, 17, 23, 0, 2, 10, 24, 23, 25, 0, 15, 30, 37, 0, 5, 0, 42, 36, 19, 7, 35, 37, 40, 16, 10, 0, 54, 43, 24, 3, 25, 47, 73, 7, 2, 20, 67, 53, 50, 0, 45, 60, 60, 0, 17, 0, 64, 73, 42, 13, 60, 67, 87, 26, 9, 0, 115, 73

Offset: 1

Seiichi Manyama, Mar 21 2017

nonn

			Divisors of 12 are 1 2 3 4 6 12.
And 2 == 12 mod 5.
We get a(12) = 2 + 3 + 12 = 17.

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

Cf. A284280, A284281.

Cf. A003106, A284150 (Sum_{d|n, d==1 or 4 mod 5} d).

a(n) = A000203(n) -5*A000203(n/5) -A284150(n), where A000203(.) =0 for non-integer arguments. - R. J. Mathar, Mar 21 2017

a(n) = A284280(n) + A284281(n). - Seiichi Manyama, Mar 24 2017

A363027 Sum of divisors of 5n-4 of form 5k+2.

Original entry on oeis.org

0, 2, 0, 2, 7, 2, 0, 14, 0, 2, 17, 9, 0, 24, 0, 2, 27, 2, 7, 46, 0, 2, 37, 2, 0, 51, 0, 19, 47, 2, 0, 66, 7, 2, 57, 24, 0, 64, 0, 9, 67, 2, 0, 113, 17, 2, 84, 2, 0, 84, 0, 34, 87, 9, 0, 106, 0, 24, 97, 39, 7, 121, 0, 2, 107, 2, 0, 175, 0, 2, 144, 2, 0, 124, 7, 49, 127, 2, 17, 168, 0, 9, 137, 86, 0, 144, 0, 2

Offset: 1

Seiichi Manyama, Jul 06 2023

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A362952, A363025, A363026.

Cf. A284280, A359244, A363032.

f:= n ->
  convert(select(t -> t mod 5 = 2, numtheory:-divisors(5*n-4)),`+`):
map(f, [$1..100]); # Robert Israel, Jul 23 2023

a[n_] := DivisorSum[5*n - 4, # &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jul 06 2023 *)
Table[Total[Select[Divisors[5n-4],Mod[#,5]==2&]],{n,90}] (* Harvey P. Dale, Feb 01 2025 *)

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

a(n) = A284280(5*n-4).

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

A363898 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(5k)).

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 7, 0, 9, 0, 8, 1, 9, 0, 10, 3, 12, 0, 14, 0, 13, 1, 18, 0, 15, 0, 17, 0, 19, 5, 21, 1, 19, 0, 20, 0, 28, 0, 24, 0, 23, 1, 28, 7, 25, 3, 27, 0, 29, 0, 36, 1, 29, 0, 35, 0, 32, 9, 34, 0, 36, 1, 38, 0, 45, 0, 43, 0, 39, 0, 38, 12, 39, 0, 40, 3, 42, 0, 63, 5, 43, 1

Offset: 1

Seiichi Manyama, Jun 27 2023

nonn

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

Cf. A363897, A363899, A363900.

Cf. A001877, A284280.

a[n_] := DivisorSum[n, # &, Mod[n/#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 27 2023 *)

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

a(n) = Sum_{d|n, n/d==2 mod 5} d.

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

A109698 Number of partitions of n into parts each congruent to 2 mod 5.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 3, 3, 4, 4, 4, 6, 4, 7, 5, 8, 7, 8, 9, 9, 10, 12, 11, 15, 12, 17, 15, 18, 19, 20, 22, 24, 24, 29, 26, 34, 31, 37, 38, 40, 44, 46, 49, 55, 53, 64, 60, 71, 71, 77, 83, 86, 93, 100, 101, 116, 112, 130, 129, 142, 149, 156, 168, 177

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(12)=2 since 12 = 12 = 2+2+2+2+2+2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284280.

g:=1/product(1-x^(2+5*i),i=0..20): gser:=series(g,x=0,86): seq(coeff(gser,x,n),n=0..82); # Emeric Deutsch, Feb 15 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+2)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Vec(prod(k=0, 100, 1/(1 - x^(5*k + 2))) + O(x^111)) \\ Indranil Ghosh, Mar 24 2017

G.f.: 1/Product_{j>=0} (1 - x^(2+5j)). - Emeric Deutsch, Feb 15 2006
a(n) ~ Gamma(2/5) * exp(Pi*sqrt(2*n/15)) / (2^(17/10) * 3^(1/5) * 5^(3/10)*Pi^(3/5) * n^(7/10)) * (1 + (11*Pi/(120*sqrt(30)) - 7*sqrt(3/10)/(5*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284280(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 24 2017

More terms from Emeric Deutsch, Feb 15 2006

A362952 Sum of divisors of 5n-1 of form 5k+2.

Original entry on oeis.org

2, 0, 9, 0, 14, 0, 19, 0, 24, 7, 29, 0, 34, 0, 39, 0, 63, 0, 49, 0, 54, 0, 59, 24, 64, 0, 69, 0, 86, 0, 108, 0, 84, 0, 89, 0, 94, 34, 99, 0, 133, 0, 109, 0, 153, 0, 119, 0, 124, 0, 129, 44, 168, 0, 139, 0, 144, 17, 198, 0, 154, 0, 159, 0, 203, 54, 169, 0, 174, 0, 179, 0, 243, 0, 228, 0, 238, 0, 199, 64, 204

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A363025, A363026, A363027.

Cf. A284280, A359287, A363028.

a[n_] := DivisorSum[5*n - 1, # &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jul 06 2023 *)

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

a(n) = A284280(5*n-1).

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

A363025 Sum of divisors of 5n-2 of form 5k+2.

Original entry on oeis.org

0, 2, 0, 2, 0, 9, 0, 2, 0, 14, 0, 2, 7, 19, 0, 2, 0, 24, 0, 9, 0, 41, 0, 2, 0, 34, 7, 2, 0, 39, 17, 2, 0, 63, 0, 2, 0, 49, 0, 24, 7, 54, 0, 2, 0, 71, 0, 26, 27, 64, 0, 2, 0, 69, 7, 2, 0, 118, 0, 2, 0, 108, 0, 2, 17, 84, 37, 2, 7, 101, 0, 2, 0, 94, 0, 78, 0, 99, 0, 2, 0, 133, 7, 24, 47, 109, 0, 2, 0, 153, 0

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A362952, A363026, A363027.

Cf. A284280, A359269, A363029.

a[n_] := DivisorSum[5*n - 2, # &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jul 06 2023 *)

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

a(n) = A284280(5*n-2).

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

A363026 Sum of divisors of 5n-3 of form 5k+2.

Original entry on oeis.org

2, 7, 14, 17, 24, 27, 34, 37, 51, 47, 54, 57, 64, 67, 86, 84, 84, 87, 94, 97, 121, 107, 121, 117, 124, 127, 168, 137, 144, 154, 154, 157, 191, 167, 174, 177, 191, 204, 238, 197, 204, 207, 214, 224, 261, 227, 234, 237, 266, 247, 315, 257, 264, 267, 291, 277, 331, 294, 294, 324, 304, 307, 378, 317, 331, 327

Offset: 1

Seiichi Manyama, Jul 06 2023

nonn

Cf. A362952, A363025, A363027.

Cf. A284280, A359237, A363030.

a[n_] := DivisorSum[5*n - 3, # &, Mod[#, 5] == 2 &]; Array[a, 100] (* Amiram Eldar, Jul 06 2023 *)

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

a(n) = A284280(5*n-3).

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

A363926 Expansion of Sum_{k>0} x^(2k) / (1 - x^(5k))^2.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 4, 0, 3, 0, 1, 4, 1, 0, 1, 2, 6, 0, 4, 0, 1, 6, 3, 0, 1, 0, 8, 0, 5, 2, 4, 8, 1, 0, 1, 0, 12, 0, 6, 0, 1, 10, 4, 2, 1, 4, 12, 0, 7, 0, 3, 12, 1, 0, 4, 0, 14, 2, 8, 0, 6, 14, 5, 0, 3, 0, 19, 0, 9, 0, 1, 18, 1, 0, 1, 6, 18, 0, 15, 4, 1, 18, 6, 0, 1, 2, 20, 0

Offset: 1

Seiichi Manyama, Jun 28 2023

nonn

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

Cf. A363925, A363928, A363929.

Cf. A001877, A284280.

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

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

a(n) = (1/5) * Sum_{d|n, d==2 mod 5} (d+3) = (3 * A001877(n) + A284280(n))/5.

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

cp's OEIS Frontend

A109091 Expansion of (1 - eta(q)^5 / eta(q^5)) / 5 in powers of q.

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A284281 a(n) = Sum_{d|n, d == 3 (mod 5)} d.

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A284152 a(n) = Sum_{d|n, d == 2 or 3 mod 5} d.

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A363027 Sum of divisors of 5*n-4 of form 5*k+2.

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A363898 Expansion of Sum_{k>0} k * x^(2*k) / (1 - x^(5*k)).

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A109698 Number of partitions of n into parts each congruent to 2 mod 5.

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A362952 Sum of divisors of 5*n-1 of form 5*k+2.

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A363025 Sum of divisors of 5*n-2 of form 5*k+2.

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A363027 Sum of divisors of 5n-4 of form 5k+2.

A363898 Expansion of Sum_{k>0} k * x^(2k) / (1 - x^(5k)).

A362952 Sum of divisors of 5n-1 of form 5k+2.

A363025 Sum of divisors of 5n-2 of form 5k+2.

A363026 Sum of divisors of 5n-3 of form 5k+2.

A363926 Expansion of Sum_{k>0} x^(2k) / (1 - x^(5k))^2.