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 1-4 of 4 results.

A356125 a(n) = Sum_{k=1..n} k * sigma_2(k).

Original entry on oeis.org

1, 11, 41, 125, 255, 555, 905, 1585, 2404, 3704, 5046, 7566, 9776, 13276, 17176, 22632, 27562, 35752, 42630, 53550, 64050, 77470, 89660, 110060, 126335, 148435, 170575, 199975, 224393, 263393, 293215, 336895, 377155, 426455, 471955, 540751, 591441, 660221, 726521
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2022

Keywords

Links

Crossrefs

Partial sums of A328259.
Column k=3 of A356124.
Cf. A319086, A356042.

Programs

  • Mathematica
    a[n_] := Sum[k * DivisorSigma[2, k], {k, 1, n}]; Array[a, 39] (* Amiram Eldar, Jul 28 2022 *)
  • PARI
    a(n) = sum(k=1, n, k*sigma(k, 2));
  • PARI
    a(n) = sum(k=1, n, k^3*binomial(n\k+1, 2));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-x^k)^2)/(1-x))
  • Python
    from math import isqrt
def A356125(n): return (-((s:=isqrt(n))*(s+1))**3>>1) + sum(k*(q:=n//k)*(q+1)*(2*k**2+q*(q+1)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^3 * binomial(floor(n/k)+1,2).
G.f.: (1/(1-x)) * Sum_{k>=1} k^3 * x^k/(1 - x^k)^2.
a(n) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Aug 02 2022

A356129 a(n) = Sum_{k=1..n} k * sigma_{n-1}(k).

Original entry on oeis.org

1, 7, 41, 395, 4503, 68969, 1205345, 24831145, 574932340, 14936279962, 427782949566, 13426887958078, 457622797727840, 16842616079514468, 665489067204502336, 28102162931539093732, 1262904299189373463930, 60182778247311758955112
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2022

Keywords

Links

Crossrefs

Cf. A356124, A356130, A356131.

Programs

  • Mathematica
    a[n_] := Sum[k * DivisorSigma[n - 1, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)
  • PARI
    a(n) = sum(k=1, n, k*sigma(k, n-1));
  • PARI
    a(n) = sum(k=1, n, k^n*binomial(n\k+1, 2));
  • Python
    from math import isqrt
from sympy import bernoulli
def A356129(n): return ((s:=isqrt(n))*(s+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))+sum(k**n*(n+1)*(q:=n//k)*(q+1)+(k*(bernoulli(n+1,q+1)-bernoulli(n+1))<<1) for k in range(1,s+1)))//(n+1)>>1 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^n * binomial(floor(n/k)+1,2).
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^n * x^k/(1 - x^k)^2.
a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Aug 07 2022

A356126 a(n) = Sum_{k=1..n} k * sigma_3(k).

Original entry on oeis.org

1, 19, 103, 395, 1025, 2537, 4945, 9625, 16438, 27778, 42430, 66958, 95532, 138876, 191796, 266692, 350230, 472864, 603204, 787164, 989436, 1253172, 1533036, 1926156, 2319931, 2834263, 3386143, 4089279, 4796589, 5749149, 6672701, 7871069, 9101837, 10605521
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2022

Keywords

Links

Crossrefs

Partial sums of A281372.
Column k=4 of A356124.
Cf. A356043.

Programs

  • Mathematica
    a[n_] := Sum[k * DivisorSigma[3, k], {k, 1, n}]; Array[a, 34] (* Amiram Eldar, Jul 28 2022 *)
  • PARI
    a(n) = sum(k=1, n, k*sigma(k, 3));
  • PARI
    a(n) = sum(k=1, n, k^4*binomial(n\k+1, 2));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k/(1-x^k)^2)/(1-x))
  • Python
    from math import isqrt
def A356126(n): return ((-(s:=isqrt(n))**2*(s+1)**2*((s<<1)+1)*(s*(3*(s+1))-1)>>1)+sum(k*(q:=n//k)*(q+1)*(15*k**3+((q<<1)+1)*(q*(3*(q+1))-1)) for k in range(1,s+1)))//30 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^4 * binomial(floor(n/k)+1,2).
G.f.: (1/(1-x)) * Sum_{k>=1} k^4 * x^k/(1 - x^k)^2.

A356128 a(n) = Sum_{k=1..n} k * sigma_n(k).

Original entry on oeis.org

1, 11, 103, 1373, 20657, 381795, 7921825, 187452793, 4916743582, 142471278944, 4506381463150, 154747691135574, 5729252807696052, 227595085199164036, 9654855890695727316, 435664037303036699736, 20836069678062430493950, 1052867409176853099312712
Offset: 1

Views

Author

Seiichi Manyama, Jul 27 2022

Keywords

Links

Crossrefs

Cf. A356046, A356124.

Programs

  • Mathematica
    a[n_] := Sum[k * DivisorSigma[n, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)
  • PARI
    a(n) = sum(k=1, n, k*sigma(k, n));
  • PARI
    a(n) = sum(k=1, n, k^(n+1)*binomial(n\k+1, 2));
  • Python
    from math import isqrt
from sympy import bernoulli
def A356128(n): return ((s:=isqrt(n))*(s+1)*(bernoulli(n+2)-bernoulli(n+2,s+1))+sum(k**(n+1)*(n+2)*(q:=n//k)*(q+1)+(k*(bernoulli(n+2,q+1)-bernoulli(n+2))<<1) for k in range(1,s+1)))//(n+2)>>1 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^(n+1) * binomial(floor(n/k)+1,2).
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^(n+1) * x^k/(1 - x^k)^2.
Showing 1-4 of 4 results.

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