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 1-3 of 3 results.

A344044 a(n) = Sum_{d|n} sigma(d)^3.

Original entry on oeis.org

1, 28, 65, 371, 217, 1820, 513, 3746, 2262, 6076, 1729, 24115, 2745, 14364, 14105, 33537, 5833, 63336, 8001, 80507, 33345, 48412, 13825, 243490, 30008, 76860, 66262, 190323, 27001, 394940, 32769, 283584, 112385, 163324, 111321, 839202, 54873, 224028, 178425, 812882, 74089
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Links

Crossrefs

Cf. A002117, A007429, A065018, A344043, A344047.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Array[a, 41] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d)^3);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^3*x^k/(1-x^k)))

Formula

G.f.: Sum_{k >= 1} sigma(k)^3 * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p+1)^3.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^10*zeta(3)/194400) * Product_{p prime} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 1.6422194986... . - Amiram Eldar, Nov 20 2022

A344042 a(n) = n * Sum_{d|n} sigma(d)^2 / d.

Original entry on oeis.org

1, 11, 19, 71, 41, 209, 71, 367, 226, 451, 155, 1349, 209, 781, 779, 1695, 341, 2486, 419, 2911, 1349, 1705, 599, 6973, 1166, 2299, 2278, 5041, 929, 8569, 1055, 7359, 2945, 3751, 2911, 16046, 1481, 4609, 3971, 15047, 1805, 14839, 1979, 11005, 9266, 6589, 2351, 32205, 3746, 12826, 6479
Offset: 1

Views

Author

Seiichi Manyama, May 08 2021

Keywords

Crossrefs

Cf. A060640, A065018, A226564, A344043.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, DivisorSigma[1, #]^2/# &]; Array[a, 51] (* Amiram Eldar, May 08 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, sigma(d)^2/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^2*x^k/(1-x^k)^2))
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - p^2*X^2) / ((1 - X) * (1 - p*X)^3 * (1 - p^2*X)))[n], ", ")) \\ Vaclav Kotesovec, May 08 2021

Formula

G.f.: Sum_{k >= 1} sigma(k)^2 * x^k/(1 - x^k)^2.
From Vaclav Kotesovec, May 08 2021: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1)^3 * zeta(s-2) / zeta(2*s-2).
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * zeta(3) * n^3 / 36. (End)

A344082 a(n) = n * Sum_{d|n} tau(d)^3 / d, where tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 10, 11, 47, 13, 110, 15, 158, 60, 130, 19, 517, 21, 150, 143, 441, 25, 600, 27, 611, 165, 190, 31, 1738, 92, 210, 244, 705, 37, 1430, 39, 1098, 209, 250, 195, 2820, 45, 270, 231, 2054, 49, 1650, 51, 893, 780, 310, 55, 4851, 132, 920, 275, 987, 61, 2440, 247, 2370, 297, 370, 67, 6721, 69
Offset: 1

Views

Author

Seiichi Manyama, May 09 2021

Keywords

Links

Crossrefs

Cf. A007429, A013661, A062369, A097988, A344043.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, DivisorSigma[0, #]^3/# &]; Array[a, 61] (* Amiram Eldar, May 09 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, numdiv(d)^3/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k)^3*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k >= 1} tau(k)^3 * x^k/(1 - x^k)^2.
If p is prime, a(p) = 8 + p.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)^4 * Product_{p prime} (1 + 4/p^2 + 1/p^4) = 31.237542262502... . - Amiram Eldar, Dec 22 2023
From Peter Bala, Jan 25 2024: (Start)
a(n) = Sum_{d|n, e|n} gcd(d, e) * tau(n/d) * tau(n/e) (the sum is a multiplicative function of n - see Tóth).
Multiplicative: a(p^k) = ( p^(k+2)*(p^2 + 4*p + 1) - p^3*(k + 2)^3 + p^2*(3*k^3 + 15*k^2 + 21*k + 5) - p*(3*k^3 + 12*k^2 + 12*k + 4) + (k + 1)^3 ) / (p - 1)^4. (End)
Showing 1-3 of 3 results.

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