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.

A350125 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^n.

Original entry on oeis.org

1, 8, 40, 345, 3303, 50225, 833569, 17045934, 388654659, 10039636255, 285508661853, 8924967326015, 302927979357701, 11114722212099135, 437913155876193839, 18447871416712820782, 827249276230172525622, 39347009369000530723017
Offset: 1

Views

Author

Seiichi Manyama, Dec 15 2021

Keywords

Links

Crossrefs

Cf. A332469, A350109, A350123, A350124, A350128.

Programs

  • Mathematica
    a[n_] := Sum[k^2 * Floor[n/k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Oct 04 2023 *)
  • PARI
    a(n) = sum(k=1, n, k^2*(n\k)^n);
  • PARI
    a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d^n-(d-1)^n)/d^2));

Formula

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d^n - (d - 1)^n)/d^2.
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) ~ n^n. - Vaclav Kotesovec, Dec 16 2021

A350107 a(n) = Sum_{k=1..n} k * floor(n/k)^2.

Original entry on oeis.org

1, 6, 14, 31, 45, 81, 101, 150, 191, 253, 285, 401, 439, 527, 623, 752, 802, 979, 1035, 1233, 1369, 1509, 1577, 1901, 2020, 2186, 2362, 2642, 2728, 3136, 3228, 3549, 3765, 3983, 4215, 4772, 4882, 5126, 5382, 5932, 6054, 6630, 6758, 7202, 7664, 7960, 8100, 8936
Offset: 1

Views

Author

Seiichi Manyama, Dec 14 2021

Keywords

Links

Crossrefs

Column 2 of A350106.
Cf. A000005 (tau), A024916, A143127, A222548, A350123, A350128.

Programs

  • Mathematica
    a[n_] := Sum[k * Floor[n/k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[2*k*DivisorSigma[0, k] - DivisorSigma[1, k], {k, 1, 100}]] (* Vaclav Kotesovec, Dec 16 2021 *)
  • PARI
    a(n) = sum(k=1, n, k*(n\k)^2);
  • PARI
    a(n) = sum(k=1, n, k*sumdiv(k, d, (2*d-1)/d));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1-x^k)^2)/(1-x))
  • PARI
    a(n) = sum(k=1, n, 2*k*numdiv(k)-sigma(k));
  • Python
    from math import isqrt
def A350107(n): return -(s:=isqrt(n))**3*(s+1)+sum((q:=n//k)*((k<<1)*((q<<1)+1)-q-1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k * Sum_{d|k} (2*d - 1)/d = 2 * A143127(n) - A024916(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^k)^2.
a(n) = Sum_{k=1..n} 2 * k * tau(k) - sigma(k).

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)

A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

Original entry on oeis.org

1, 16, 62, 219, 405, 1053, 1523, 2948, 4407, 7041, 8703, 15283, 17949, 24657, 32685, 44806, 50536, 70687, 78573, 105411, 125879, 149879, 163565, 222425, 247476, 286134, 327634, 396258, 423084, 532236, 564818, 664763, 738095, 821693, 904937, 1107618, 1162268, 1277588, 1395760
Offset: 1

Views

Author

Seiichi Manyama, Jul 31 2022

Keywords

Links

Crossrefs

Column k=3 of A356250.
Cf. A000537, A024916, A318742, A350123.
Cf. A064603, A356125, A319086.

Programs

  • Mathematica
    a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jul 31 2022 *)
  • PARI
    a(n) = sum(k=1, n, (k*(n\k))^3);
  • PARI
    a(n) = sum(k=1, n, k^3*sumdiv(k, d, 1-(1-1/d)^3));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))
  • Python
    from math import isqrt
def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^3 * Sum_{d|k} (1 - (1 - 1/d)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
From Vaclav Kotesovec, Aug 02 2022: (Start)
a(n) = A064603(n) - 3*A356125(n) + 3*A319086(n).
a(n) ~ n^4 * (Pi^2/8 + Pi^4/360 - 3*zeta(3)/4). (End)

A356250 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (j * floor(n/j))^k.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 22, 15, 5, 1, 32, 62, 57, 21, 6, 1, 64, 178, 219, 91, 33, 7, 1, 128, 518, 849, 405, 185, 41, 8, 1, 256, 1522, 3315, 1843, 1053, 247, 56, 9, 1, 512, 4502, 13017, 8541, 6065, 1523, 402, 69, 10, 1, 1024, 13378, 51339, 40171, 35253, 9571, 2948, 545, 87, 11
Offset: 1

Views

Author

Seiichi Manyama, Jul 31 2022

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1,      1, ...
  2,  4,   8,   16,   32,    64,    128, ...
  3,  8,  22,   62,  178,   518,   1522, ...
  4, 15,  57,  219,  849,  3315,  13017, ...
  5, 21,  91,  405, 1843,  8541,  40171, ...
  6, 33, 185, 1053, 6065, 35253, 206345, ...
  7, 41, 247, 1523, 9571, 61091, 394987, ...

Crossrefs

Columns k=0..3 give A001477, A024916, A350123, A356249.
T(n,n) gives A356238.
Cf. A344725.

Programs

  • Mathematica
    T[n_, k_] := Sum[(j * Floor[n/j])^k, {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 31 2022 *)
  • PARI
    T(n, k) = sum(j=1, n, (j*(n\j))^k);
  • PARI
    T(n, k) = if(k==0, n, sum(j=1, n, j^k*sumdiv(j, d, 1-(1-1/d)^k)));

Formula

T(n,k) = Sum_{j=1..n} j^k * Sum_{d|j} (1 - (1 - 1/d)^k) for k > 0.
Showing 1-5 of 5 results.

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