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.

Previous Showing 11-16 of 16 results.

A350108 a(n) = Sum_{k=1..n} k * floor(n/k)^3.

Original entry on oeis.org

1, 10, 32, 87, 153, 309, 443, 722, 1005, 1443, 1785, 2605, 3087, 3951, 4875, 6154, 6988, 8809, 9855, 12057, 13853, 16001, 17543, 21347, 23478, 26484, 29440, 33696, 36162, 41994, 44816, 50351, 54755, 59909, 64577, 73524, 77558, 84002, 90142, 100072, 105034
Offset: 1

Views

Author

Seiichi Manyama, Dec 14 2021

Keywords

Links

Crossrefs

Column 3 of A350106.
Cf. A024916, A143128, A143127, A318742.

Programs

  • Mathematica
    a[n_] := Sum[k * Floor[n/k]^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[(1 + 3*k)*DivisorSigma[1, k] - 3*k*DivisorSigma[0, k], {k, 1, 50}]] (* Vaclav Kotesovec, Dec 16 2021 *)
  • PARI
    a(n) = sum(k=1, n, k*(n\k)^3);
  • PARI
    a(n) = sum(k=1, n, k*sumdiv(k, d, (d^3-(d-1)^3)/d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1-x^k)^2)/(1-x))
  • Python
    from math import isqrt
def A350108(n): return -(s:=isqrt(n))**4*(s+1)+sum((q:=n//k)*(k**2*(3*(q+1))+k*(q*((q<<1)-3)-3)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 31 2023

Formula

a(n) = Sum_{k=1..n} k * Sum_{d|k} (d^3 - (d - 1)^3)/d.
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 - x^k)^2.
From Vaclav Kotesovec, Aug 03 2022: (Start)
a(n) = A024916(n) + 3*A143128(n) - 3*A143127(n).
a(n) ~ Pi^2*n^3/6 - 3*n^2*log(n)/2. (End)

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)

A110662 Triangle read by rows: T(n,k) is the sum of the sums of divisors of k, k+1, ..., n (1 <= k <= n).

Original entry on oeis.org

1, 4, 3, 8, 7, 4, 15, 14, 11, 7, 21, 20, 17, 13, 6, 33, 32, 29, 25, 18, 12, 41, 40, 37, 33, 26, 20, 8, 56, 55, 52, 48, 41, 35, 23, 15, 69, 68, 65, 61, 54, 48, 36, 28, 13, 87, 86, 83, 79, 72, 66, 54, 46, 31, 18, 99, 98, 95, 91, 84, 78, 66, 58, 43, 30, 12, 127, 126, 123, 119, 112
Offset: 1

Views

Author

Emeric Deutsch, Aug 02 2005

Keywords

Comments

Equals A000012 * (A000203 * 0^(n-k)) * A000012, 1 <= k <= n. - Gary W. Adamson, Jul 26 2008
Row sums = A143128. - Gary W. Adamson, Jul 26 2008

Examples

			T(4,2)=14 because the divisors of 2 are {1,2}, the divisors of 3 are {1,3} and the divisors of 4 are {1,2,4}; sum of all these divisors is 14.
Triangle begins:
   1;
   4,  3;
   8,  7,  4;
  15, 14, 11,  7;
  21, 20, 17, 13,  6;
  ...

Links

Crossrefs

Cf. A000203, A024916.
Cf. A143128.

Programs

  • Maple
    with(numtheory): T:=(n,k)->add(sigma(j),j=k..n): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
  • Mathematica
    T[n_, n_] := DivisorSigma[1, n]; T[n_, k_] := Sum[DivisorSigma[1, j], {j, k, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 03 2017 *)

Formula

T(n, k) = Sum_{j=k..n} sigma(j), where sigma(j) is the sum of the divisors of j.
T(n, n) = sigma(n) = A000203(n) = sum of divisors of n.
T(n, 1) = Sum_{j=1..n} sigma(j) = A024916(n).

A245100 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(n), multiplied by n.

Original entry on oeis.org

1, 6, 6, 6, 28, 15, 15, 72, 28, 28, 120, 45, 27, 45, 90, 90, 66, 66, 336, 91, 91, 168, 168, 120, 120, 120, 496, 153, 153, 702, 190, 190, 840, 231, 105, 105, 231, 396, 396, 276, 276, 1440, 325, 125, 325, 546, 546, 378, 162, 162, 378, 1568, 435, 435, 2160, 496, 496, 2016
Offset: 1

Views

Author

Omar E. Pol, Jul 11 2014

Keywords

Comments

Row sums give A064987.
Since both A000203(n) and A024916(n) have a symmetric representation then both row n and the triangle have can be represented as a symmetric polycube.

Examples

			The irregular triangle begins:
1;
6;
6, 6;
28;
15, 15;
72;
28, 28;
120;
45, 27, 45;
90, 90;
66, 66;
336;
91, 91;
168, 168;
120, 120, 120;
496;
153, 153;
702;
190, 190;
840;
231, 105, 105, 231;
...
For n = 9 the parts of the symmetric representation of sigma(9) are [5, 3, 5], so row 9 is [45, 27, 45].

Crossrefs

Cf. A000203, A024916, A064987, A143128, A185283, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A244580.

Formula

T(n,k) = n*A237270(n,k).

A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

Original entry on oeis.org

1, 25, 133, 581, 1331, 3923, 6667, 14347, 23824, 41824, 57796, 106180, 136938, 202794, 283794, 410770, 499204, 726652, 863832, 1199832, 1496184, 1879512, 2171520, 3000960, 3485335, 4223527, 5010847, 6240159, 6971829, 8915829, 9869141, 11933525, 13658501
Offset: 1

Views

Author

Seiichi Manyama, Oct 20 2023

Keywords

Links

Crossrefs

Partial sums of A282211.
Cf. A319086, A320895, A356125, A364269.
Cf. A000203, A000537, A143128.

Programs

  • Mathematica
    Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)
  • PARI
    f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=3, t=1) = sum(k=1, n, k^(s+t)*f(n\k, s));
  • Python
    def A364194(n): return sum((k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023
  • Python
    from math import isqrt
def A364194(n): return ((((s:=isqrt(n))*(s + 1))**3*(2*s+1)*(1-3*s*(s+1))>>1) + sum((q:=n//k)*(q+1)*k**3*(q*(15*k+q*(15*k+12*q+18)+2)-2) for k in range(1,s+1)))//60 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^4 * A000537(floor(n/k)).
a(n) ~ (zeta(2)/5) * n^5. - Amiram Eldar, Oct 20 2023

A321675 a(n) = Sum_{k=1..10^n} k*sigma(k).

Original entry on oeis.org

1, 622, 558275, 549175530, 548429473046, 548320905633448, 548312690631798482, 548311465139943768941, 548311366911386862908968, 548311356554322895313137239, 548311355740964925044531454428, 548311355626818302486560961291870, 548311355617569600726982364186141942
Offset: 0

Views

Author

Daniel Suteu, Nov 16 2018

Keywords

Crossrefs

Cf. A000203, A064987, A072692, A086463 (Pi^2/18), A143128.

Programs

  • Mathematica
    Array[Sum[k DivisorSigma[1, k], {k, 10^#}] &, 7, 0] (* Michael De Vlieger, Nov 20 2018 *)
  • PARI
    a(n) = sum(k=1, 10^n, k*sigma(k)); \\ Michel Marcus, Nov 23 2018

Formula

a(n) = A143128(10^n).
a(n) ~ 10^(3*n) * Pi^2 / 18.
Previous Showing 11-16 of 16 results.

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