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.

A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.

Original entry on oeis.org

1, 8, 22, 57, 91, 185, 247, 402, 545, 775, 917, 1379, 1573, 1995, 2455, 3106, 3428, 4377, 4775, 5909, 6753, 7727, 8301, 10331, 11230, 12564, 13904, 15990, 16888, 19908, 20930, 23597, 25545, 27767, 29827, 34468, 35910, 38660, 41328, 46318, 48080, 53644, 55578
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A064602, A143128, A222548, A350107, A350124, A350125, A350128.

Programs

  • Mathematica
    Accumulate[Table[2*k*DivisorSigma[1, k] - DivisorSigma[2, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^2*(n\k)^2);
  • PARI
    a(n) = sum(k=1, n, k^2*sumdiv(k, d, (2*d-1)/d^2));
  • PARI
    a(n) = sum(k=1, n, 2*k*sigma(k)-sigma(k, 2));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))
  • Python
    from math import isqrt
def A350123(n): return (-(s:=isqrt(n))**3*(s+1)*((s<<1)+1)+sum((q:=n//k)*(6*k**2*q+((k<<1)-1)*(q+1)*((q<<1)+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (2*d - 1)/d^2 = Sum_{k=1..n} 2 * k * sigma(k) - sigma_2(k) = 2 * A143128(n) - A064602(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) ~ n^3 * (Pi^2/9 - zeta(3)/3). - Vaclav Kotesovec, Dec 16 2021

A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.

Original entry on oeis.org

1, 8, 44, 417, 4545, 69905, 1207937, 24904806, 575256641, 14947281595, 427836523971, 13429362462839, 457637290140469, 16843379604615375, 665494379869134005, 28102480944522059434, 1262906802939553227382, 60182948301301262753877
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A222548, A319194, A350107, A350125.

Programs

  • Mathematica
    Table[Sum[k^n Floor[n/k]^2,{k,n}],{n,20}] (* Harvey P. Dale, Feb 11 2022 *)
  • PARI
    a(n) = sum(k=1, n, k^n*(n\k)^2);
  • PARI
    a(n) = sum(k=1, n, 2*k*sigma(k, n-1)-sigma(k, n));
  • Python
    from math import isqrt
from sympy import bernoulli
def A350128(n): return (((s:=isqrt(n))+1)*(1-s)*(bernoulli(n+1,s+1)-(b:=bernoulli(n+1)))+sum(k**n*(n+1)*(((q:=n//k)+1)*(q-1))+(1-2*k)*(b-bernoulli(n+1,q+1)) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} 2 * k * sigma_{n-1}(k) - sigma_{n}(k).
a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Dec 16 2021

A350124 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^3.

Original entry on oeis.org

1, 12, 40, 121, 207, 473, 649, 1142, 1611, 2401, 2853, 4647, 5285, 6879, 8759, 11452, 12558, 16739, 18127, 23353, 27129, 31171, 33219, 43573, 47524, 53210, 59538, 69996, 73274, 89694, 93446, 107195, 116731, 126545, 137505, 164580, 169946, 182244, 195644, 225454
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A318742, A350108, A350123, A350125.
Cf. A064602, A143128, A319085.

Programs

  • Mathematica
    Accumulate[Table[DivisorSigma[2, k] - 3*k*DivisorSigma[1, k] + 3*k^2*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 17 2021 *)
  • PARI
    a(n) = sum(k=1, n, k^2*(n\k)^3);
  • PARI
    a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^3-(d-1)^3)/d^2));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))
  • Python
    from math import isqrt
def A350124(n): return (-(s:=isqrt(n))**4*(s+1)*(2*s+1) + sum((q:=n//k)*(k*(3*(k-1))+q*(k*(9*(k-1))+q*(k*(12*k-6)+2)+3)+1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^3 - (d - 1)^3)/d^2.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + x^k)/(1 - x^k)^3.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A064602(n) - 3*A143128(n) + 3*A319085(n).
a(n) ~ n^3 * (log(n) + 2*gamma + (zeta(3) - 1)/3 - Pi^2/6), where gamma is the Euler-Mascheroni constant A001620. (End)

A356238 a(n) = Sum_{k=1..n} (k * floor(n/k))^n.

Original entry on oeis.org

1, 8, 62, 849, 8541, 206345, 2581403, 76623522, 1617299079, 49463993875, 952905453423, 59000021366675, 1198427462876421, 54128102218676115, 2321105129608323165, 117387839988330848902, 3205342976298888473968, 268263812478494295219717
Offset: 1

Views

Author

Seiichi Manyama, Jul 30 2022

Keywords

Links

Crossrefs

Cf. A007778, A350109, A350125, A356239, A356240.

Programs

  • Mathematica
    a[n_] := Sum[(k * Floor[n/k])^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)
  • PARI
    a(n) = sum(k=1, n, (k*(n\k))^n);
  • PARI
    a(n) = sum(k=1, n, k^n*sumdiv(k, d, 1-(1-1/d)^n));

Formula

a(n) = Sum_{k=1..n} k^n * Sum_{d|k} (1 - (1 - 1/d)^n).

A356244 a(n) = Sum_{k=1..n} (k-1)^n * Sum_{j=1..floor(n/k)} j^2.

Original entry on oeis.org

0, 1, 9, 102, 1304, 20784, 377286, 7934693, 186969913, 4918785791, 142381832107, 4506907611825, 154723950495961, 5729421493899419, 227586600129484543, 9654927881195999544, 435660032125475809618, 20836109197604840372979, 1052865018045922422499409
Offset: 1

Views

Author

Seiichi Manyama, Jul 30 2022

Keywords

Crossrefs

Cf. A000330, A350125, A356131, A356243.

Programs

  • Mathematica
    a[n_] := Sum[(k - 1)^n * Sum[j^2, {j, 1, Floor[n/k]}], {k, 1, n}]; Array[a, 19] (* Amiram Eldar, Jul 30 2022 *)
  • PARI
    a(n) = sum(k=1, n, (k-1)^n*sum(j=1, n\k, j^2));
  • PARI
    a(n) = sum(k=1, n, k^2*(sigma(k, n-2)-(n\k)^n));
  • PARI
    a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d-1)^n/d^2));

Formula

a(n) = Sum_{k=1..n} (k-1)^n * A000330(floor(n/k)).
a(n) = Sum_{k=1..n} k^2 * (sigma_{n-2}(k) - floor(n/k)^n) = A356243(n) - A350125(n).
a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d - 1)^n / d^2.
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k * (1 + x^k)/(1 - x^k)^3.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.