A284103 -id:A284103 - 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

A284105 a(n) = Sum_{d|n, d == 6 (mod 7)} d.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 13, 0, 0, 0, 0, 6, 0, 20, 0, 0, 0, 6, 0, 13, 27, 0, 0, 6, 0, 0, 0, 34, 0, 6, 0, 0, 13, 20, 41, 6, 0, 0, 0, 0, 0, 54, 0, 0, 0, 13, 0, 33, 55, 0, 0, 0, 0, 26, 0, 62, 0, 0, 13, 6, 0, 34, 69, 0, 0, 6, 0, 0, 0, 76, 0, 19, 0, 20, 27, 41

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A109708.
Cf. Sum_{d|n, d == k-1 (mod k)} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), A284103 (k=5), A284104 (k=6), this sequence (k=7).
Cf. Sum_{d|n, d == k (mod 7)} d: A284099 (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), this sequence (k=6).

Table[Sum[If[Mod[d,7] == 6,d, 0], {d, Divisors[n]}], {n, 82}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d,7)==6, d, 0)),", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%7==6]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (7*k - 1)*x^(7*k-1)/(1 - x^(7*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023

A284280 a(n) = Sum_{d|n, d == 2 (mod 5)} d.

Original entry on oeis.org

0, 2, 0, 2, 0, 2, 7, 2, 0, 2, 0, 14, 0, 9, 0, 2, 17, 2, 0, 2, 7, 24, 0, 14, 0, 2, 27, 9, 0, 2, 0, 34, 0, 19, 7, 14, 37, 2, 0, 2, 0, 51, 0, 24, 0, 2, 47, 14, 7, 2, 17, 54, 0, 29, 0, 9, 57, 2, 0, 14, 0, 64, 7, 34, 0, 24, 67, 19, 0, 9, 0, 86, 0, 39, 0, 2, 84, 2, 0, 2

Offset: 1

Seiichi Manyama, Mar 24 2017

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

Cf. A013661, A109698, A284152.

Cf. Sum_{d|n, d=k mod 5} d: A284097 (k=1), this sequence (k=2), A284281 (k=3), A284103 (k=4).

Table[Sum[If[Mod[d, 5] == 2, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 24 2017 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d, 5)==2, 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==2]) # Indranil Ghosh, Mar 24 2017

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

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

A109700 Number of partitions of n into parts each equal to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 3, 4, 2, 2, 3, 5, 4, 3, 3, 6, 6, 6, 4, 6, 7, 9, 7, 7, 8, 11, 11, 11, 9, 12, 14, 16, 13, 14, 16, 21, 20, 19, 18, 24, 26, 27, 24, 27, 31, 36, 34, 34, 35, 43, 45, 47, 43, 49, 55, 62, 58, 59, 63, 75, 77, 77, 75, 87

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(30)=2 since 30 = 14+4+4+4+4 = 9+9+4+4+4

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

Cf. A284103.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), this sequence (m=5), A109702 (m=6), A109708 (m=7).

g:=1/product(1-x^(4+5*j),j=0..25): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..90); # Emeric Deutsch, Mar 30 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+4)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(4+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(4/5) * exp(Pi*sqrt(2*n/15)) / (2^(19/10) * 3^(2/5) * 5^(1/10) * Pi^(1/5) * n^(9/10)) * (1 - (9*sqrt(6/5)/(5*Pi) + Pi/(120*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

More terms from Michael Somos, Aug 10 2005

A284317 Expansion of Product_{k>=0} (1 - x^(5*k+4)) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 3, -1, 0, 0, -2, 3, -1, 0, 0, -3, 4, -1, 0, 1, -4, 4, -1, 0, 1, -5, 5, -1, 0, 2, -7, 5, -1, 0, 3, -8, 6, -1, 0, 5, -10, 6, -1, -1, 6, -12, 7, -1, -1, 9, -14

Offset: 0

Seiichi Manyama, Mar 25 2017

sign

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), A284315 (m=3), A284316 (m=4), this sequence (m=5).

Cf. A281243, A284103.

S:= series(mul(1-x^(5*k+4),k=0..200),x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 27 2017

CoefficientList[Series[Product[1 - x^(5k + 4), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(5*k + 4)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1.

G.f. is the QPochhammer symbol (x^4;x^5)infinity. - _Robert Israel, Mar 27 2017

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.

A284104 a(n) = Sum_{d|n, d == 5 (mod 6)} d.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 0, 0, 0, 5, 11, 0, 0, 0, 5, 0, 17, 0, 0, 5, 0, 11, 23, 0, 5, 0, 0, 0, 29, 5, 0, 0, 11, 17, 40, 0, 0, 0, 0, 5, 41, 0, 0, 11, 5, 23, 47, 0, 0, 5, 17, 0, 53, 0, 16, 0, 0, 29, 59, 5, 0, 0, 0, 0, 70, 11, 0, 17, 23, 40, 71, 0, 0, 0, 5, 0, 88, 0, 0, 5

Offset: 1

Seiichi Manyama, Mar 20 2017

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

Cf. A086729, A109702.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), A050452 (k=4), A284103 (k=5), this sequence (k=6), A284105 (k=7).

Table[Sum[If[Mod[d, 6] == 5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)
Table[Total[Select[Divisors[n],Mod[#,6]==5&]],{n,80}] (* Harvey P. Dale, Dec 30 2017 *)

for(n=1, 80, print1(sumdiv(n, d, if(Mod(d,6)==5, d, 0)),", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%6==5]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=1} (6*k - 1)*x^(6*k-1)/(1 - x^(6*k-1)). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - Amiram Eldar, Nov 26 2023

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

Original entry on oeis.org

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

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)).

cp's OEIS Frontend

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

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A284105 a(n) = Sum_{d|n, d == 6 (mod 7)} d.

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A284280 a(n) = Sum_{d|n, d == 2 (mod 5)} d.

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

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A109700 Number of partitions of n into parts each equal to 4 mod 5.

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A284317 Expansion of Product_{k>=0} (1 - x^(5*k+4)) in powers of x.

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

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

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