cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.

Original entry on oeis.org

1, 3, 3, 7, 6, 9, 8, 15, 9, 18, 12, 21, 14, 24, 18, 31, 18, 27, 20, 42, 24, 36, 24, 45, 31, 42, 27, 56, 30, 54, 32, 63, 36, 54, 48, 63, 38, 60, 42, 90, 42, 72, 44, 84, 54, 72, 48, 93, 57, 93, 54, 98, 54, 81, 72, 120, 60, 90, 60, 126, 62, 96, 72, 127, 84, 108, 68, 126, 72, 144
Offset: 1

Views

Author

Vladeta Jovovic, Dec 18 2002

Keywords

Links

Crossrefs

Cf. A000203, A035191, A046913, A346933.
Cf. A002131 (k=2), this sequence (k=3), A285895 (k=4), A285896 (k=5).
Cf. A326399, A326400.

Programs

  • Mathematica
    f[p_, e_] := If[p == 3, 3^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)
  • PARI
    for(n=1,70,d=divisors(n); s=0; for(j=1,matsize(d)[2],if((n/d[j])%3>0,s=s+d[j])); print1(s,","))
  • PARI
    a(n)=sumdiv(n,d,if((n/d)%3,1,0)*d)

Formula

G.f.: Sum_{k>0} x^k*(1+x^k)^2*(1+x^(2*k))/(1-x^(3*k))^2.
a(n) = (A000203(3*n)-A000203(n))/3. - Vladeta Jovovic, Dec 22 2003
G.f.: Sum_{k>=1} k*x^k*(1 + x^k)/(1 - x^(3*k)). - Ilya Gutkovskiy, Sep 13 2019
From R. J. Mathar, May 25 2020: (Start)
a(n) = A326399(n) + A326400(n).
a(n) = A000203(n) - A000203(n/3), where A000203(.) = 0 for non-integer arguments. (End)
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(3^e) = 3^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/27 = 0.731081... (A346933). (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^s). - Amiram Eldar, Dec 30 2022

Extensions

Extended by Klaus Brockhaus and Benoit Cloitre, Dec 20 2002

A083703 Expansion of eta(q)^4/eta(q^4) in powers of q.

Original entry on oeis.org

1, -4, 2, 8, -4, -8, -8, 16, 6, -12, 8, 8, -8, -24, 0, 16, 12, -16, 10, 24, -8, -16, -24, 16, 8, -28, 8, 32, -16, -8, 0, 32, 6, -32, 16, 16, -12, -40, -24, 16, 24, -16, 16, 40, -8, -40, 0, 32, 24, -36, 10, 16, -24, -24, -32, 48, 0, -32, 24, 24, -16, -40, 0, 48, 12, -16, 16, 56, -16, -32, -48, 16, 30, -64, 8, 40, -24
Offset: 0

Views

Author

Michael Somos, May 04 2003

Keywords

Comments

Euler transform of period 4 sequence [ -4,-4,-4,-3,...].

Links

Crossrefs

A080965(n)=(-1)^n a(n). a(2n)=0 iff n in A004215 (checked up to n=343).
a(2n)=0 iff A005875(n)=0.
Cf. A274327, A285895.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 4)=0, -3, -4), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 07 2017
  • Mathematica
    CoefficientList[QPochhammer[x]^4/QPochhammer[x^4] + O[x]^80, x] (* Jean-François Alcover, Sep 19 2016 *)
  • PARI
    a(n)=if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X)^4/eta(X^4),n))

Formula

G.f.: Product_{n>0} (1-x^n)^4/(1-x^(4n)).
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 104, 248, 560, 1200, 2474, 4924, 9520, 17928, 33008, 59528, 105408, 183536, 314744, 532208, 888382, 1465208, 2389808, 3857456, 6166096, 9766576, 15336816, 23888844, 36924656, 56659296, 86341664, 130710104, 196640576, 294059872, 437232746, 646561792
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2016

Keywords

Examples

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 104*x^4 + 248*x^5 + 560*x^6 + ...

Links

Crossrefs

Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), this sequence (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).
Cf. A083703.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^4, x^4]/QPochhammer[x, x]^4 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
  • PARI
    first(n)=my(x='x);Vec(prod(k=1,n,(1-x^(4*k))/(1-x^k)^4,1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

Formula

G.f.: Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4.
a(n) ~ 5*exp(Pi*sqrt(5*n/2)) / (2^(13/2) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^4; x^4)inf/((x; x)_inf)^4, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

A285896 Sum of divisors d of n such that n/d is not congruent to 0 mod 5.

Original entry on oeis.org

1, 3, 4, 7, 5, 12, 8, 15, 13, 15, 12, 28, 14, 24, 20, 31, 18, 39, 20, 35, 32, 36, 24, 60, 25, 42, 40, 56, 30, 60, 32, 63, 48, 54, 40, 91, 38, 60, 56, 75, 42, 96, 44, 84, 65, 72, 48, 124, 57, 75, 72, 98, 54, 120, 60, 120, 80, 90, 60, 140, 62, 96, 104, 127, 70, 144
Offset: 1

Views

Author

Seiichi Manyama, Apr 28 2017

Keywords

Examples

			The divisors of 10 are 1, 2, 5, and 10. 10/1 == 0 (mod 5) and 10/2 == 0 (mod 5). Hence, a(10) = 5 + 10 = 15.

Links

Crossrefs

Cf. A002131 (k=2), A078708 (k=3), A285895 (k=4), this sequence (k=5).
Cf. A000203.

Programs

  • Mathematica
    f[p_, e_] := If[p == 5, 5^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)
  • PARI
    a(n)=sumdiv(n, d, if(n/d%5, d, 0)); \\ Andrew Howroyd, Jul 20 2018

Formula

a(n) = (A000203(5*n)-A000203(n))/5.
G.f.: Sum_{k>=1} k*x^k*(1 + x^k + x^(2*k) + x^(3*k))/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 12 2019
From Amiram Eldar, Oct 30 2022: (Start)
Multiplicative with a(5^e) = 5^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 5.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/25 = 0.789568... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/5^s). - Amiram Eldar, Dec 30 2022

Extensions

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A327095 Expansion of Sum_{k>=1} k * x^k * (1 - x^k + x^(2*k)) / (1 - x^(4*k)).

Original entry on oeis.org

1, 1, 4, 2, 6, 4, 8, 4, 13, 6, 12, 8, 14, 8, 24, 8, 18, 13, 20, 12, 32, 12, 24, 16, 31, 14, 40, 16, 30, 24, 32, 16, 48, 18, 48, 26, 38, 20, 56, 24, 42, 32, 44, 24, 78, 24, 48, 32, 57, 31, 72, 28, 54, 40, 72, 32, 80, 30, 60, 48, 62, 32, 104, 32, 84, 48
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 13 2019

Keywords

Links

Crossrefs

Cf. A002131, A115607, A285895, A316631 (Moebius transform).

Programs

  • Maple
    N:= 100:
G:= add(k * x^k * (1 - x^k + x^(2*k)) / (1 - x^(4*k)),k=1..N):
S:= series(G,x,N+1):
[seq(coeff(S,x,i),i=1..N)];# Robert Israel, Sep 17 2019
  • Mathematica
    nmax = 66; CoefficientList[Series[Sum[k x^k (1 - x^k + x^(2 k))/(1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
A002131[n_] := Total[Select[Divisors[n], OddQ[n/#] &]]; a[n_] := If[OddQ[n], A002131[n], A002131[n] - A002131[n/2]]; Table[a[n], {n, 1, 66}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); f[2, e_] := 2^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)
  • PARI
    a(n)={sumdiv(n, d, d*((n/d%2==1) - (n/d%4==2)))} \\ Andrew Howroyd, Sep 13 2019

Formula

G.f.: Sum_{k>=1} x^k * (1 + 3 * x^(2*k) + x^(3*k) + 3 * x^(4*k) + x^(6*k)) / (1 - x^(4*k))^2.
a(n) = Sum_{d|n, n/d odd} d - Sum_{d|n, n/d twice odd} d.
a(n) = A002131(n) if n odd, A002131(n) - A002131(n/2) if n even.
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(2^e) = 2^(e-1), and a(p^e) = (p^(e+1)-1)/(p-1) if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/64 = 0.462637... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/2^s)^2. - Amiram Eldar, Jan 06 2023
Showing 1-5 of 5 results.

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