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 1-7 of 7 results.

A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 10, 1, 33, 83, 74, 32, 14, 1, 65, 245, 274, 136, 52, 16, 1, 129, 731, 1058, 644, 254, 66, 20, 1, 257, 2189, 4162, 3160, 1396, 382, 92, 23, 1, 513, 6563, 16514, 15692, 8054, 2502, 596, 115, 27, 1, 1025, 19685, 65794, 78256, 47452, 17086, 4388, 833, 147, 29
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Examples

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   5, 11,  29,   83,  245,   731, ...
   8, 22,  74,  274, 1058,  4162, ...
  10, 32, 136,  644, 3160, 15692, ...
  14, 52, 254, 1396, 8054, 47452, ...

Links

Crossrefs

Columns k=1..5 give A006218, A222548, A318742, A318743, A318744.
T(n,n) gives A332469.
Cf. A306533, A344726.

Programs

  • Mathematica
    T[n_, k_] := Sum[Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, (n\j)^k);
  • PARI
    T(n, k) = sum(j=1, n, sumdiv(j, d, d^k-(d-1)^k));
  • Python
    from math import isqrt
from itertools import count, islice
def A344725_T(n,k): return -(s:=isqrt(n))**(k+1)+sum((q:=n//w)*(w**k-(w-1)**k+q**(k-1)) for w in range(1,s+1))
def A344725_gen(): # generator of terms
     return (A344725_T(k+1,n-k) for n in count(1) for k in range(n))
A344725_list = list(islice(A344725_gen(),30)) # Chai Wah Wu, Oct 26 2023

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j).
T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k - (d - 1)^k.

A344720 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^2.

Original entry on oeis.org

1, 3, 9, 12, 22, 30, 44, 48, 71, 83, 105, 115, 141, 157, 201, 206, 240, 266, 304, 318, 378, 402, 448, 460, 519, 547, 623, 641, 699, 747, 809, 815, 907, 943, 1035, 1064, 1138, 1178, 1286, 1302, 1384, 1448, 1534, 1560, 1710, 1758, 1852, 1866, 1977, 2039, 2179, 2209, 2315, 2395, 2535
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Links

Crossrefs

Column k=2 of A344726.
Cf. A059851, A078471, A222548.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-2*DivisorSigma[0, 2*n] + 3*DivisorSigma[0, n] + 2*DivisorSigma[1, 2*n] - 4*DivisorSigma[1, n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^2);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(2*d-1)));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1+x^k))/(1-x))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (2*d - 1).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 + x^k).
a(n) ~ Pi^2 * n^2 / 12. - Vaclav Kotesovec, May 28 2021

A344721 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^3.

Original entry on oeis.org

1, 7, 27, 56, 118, 196, 324, 448, 685, 901, 1233, 1549, 2019, 2445, 3157, 3664, 4482, 5262, 6290, 7128, 8536, 9598, 11118, 12392, 14255, 15743, 18087, 19711, 22149, 24417, 27209, 29251, 32771, 35327, 39087, 42048, 46046, 49244, 54180, 57512, 62434, 66838, 72258, 76246
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Links

Crossrefs

Column k=3 of A344726.
Cf. A318742.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 6*DivisorSigma[1, n] - 3*DivisorSigma[1, 2*n] - 9/2 * DivisorSigma[2, n] + 3/2 * DivisorSigma[2, 2*n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^3);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^3-(d-1)^3)));
  • 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))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^3 - (d - 1)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k/(1 + x^k).
a(n) ~ 3*zeta(3)*n^3/4. - Vaclav Kotesovec, May 28 2021

A344724 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^n.

Original entry on oeis.org

1, 3, 27, 240, 3094, 45990, 821484, 16711680, 387177517, 9990293423, 285263019633, 8913939911695, 302862111412779, 11111328866154037, 437889173336927557, 18446462747068745474, 827238010832411671962, 39346258082152478030126
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Links

Crossrefs

Main diagonal of A344726.
Cf. A332469.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, May 27 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^n);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^n-(d-1)^n)));

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^n - (d - 1)^n).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k/(1 + x^k).
a(n) ~ n^n. - Vaclav Kotesovec, May 28 2021

A350161 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/(2*j-1))^k.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 8, 8, 3, 1, 16, 26, 15, 5, 1, 32, 80, 63, 25, 5, 1, 64, 242, 255, 125, 33, 5, 1, 128, 728, 1023, 625, 209, 45, 6, 1, 256, 2186, 4095, 3125, 1281, 335, 60, 7, 1, 512, 6560, 16383, 15625, 7745, 2385, 504, 73, 9, 1, 1024, 19682, 65535, 78125, 46593, 16775, 4080, 703, 95, 9
Offset: 1

Views

Author

Seiichi Manyama, Dec 18 2021

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,     1,      1,      1, ...
  2,  4,   8,   16,    32,     64,    128, ...
  2,  8,  26,   80,   242,    728,   2186, ...
  3, 15,  63,  255,  1023,   4095,  16383, ...
  5, 25, 125,  625,  3125,  15625,  78125, ...
  5, 33, 209, 1281,  7745,  46593, 279809, ...
  5, 45, 335, 2385, 16775, 117585, 823415, ...

Crossrefs

Columns k=1..3 give A014200, A350162, A350163.
T(n,n) gives A350164.
Cf. A101455, A344726, A350122.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^(j + 1) * Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 18 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\(2*j-1))^k);
  • PARI
    T(n, k) = sum(j=1, n, sumdiv(j, d, kronecker(-4, j/d)*(d^k-(d-1)^k)));

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 + x^(2*j)).
T(n,k) = Sum_{j=1..n} Sum_{d|j} A101455(j/d) * (d^k - (d - 1)^k).

A344722 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^4.

Original entry on oeis.org

1, 15, 81, 240, 610, 1230, 2336, 3840, 6371, 9455, 14097, 19615, 27441, 36205, 48849, 61874, 79860, 99470, 124816, 150846, 186498, 221646, 267232, 313840, 373059, 431599, 508595, 581009, 673635, 767835, 881357, 989615, 1131667, 1264111, 1429875, 1590464, 1785010
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Links

Crossrefs

Column k=4 of A344726.
Cf. A318743.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[3*DivisorSigma[0, n] - 2*DivisorSigma[0, 2*n] - 8*DivisorSigma[1, n] + 4*DivisorSigma[1, 2*n] + 9*DivisorSigma[2, n] - 3*DivisorSigma[2, 2*n] - 5*DivisorSigma[3, n] + DivisorSigma[3, 2*n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^4);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^4-(d-1)^4)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^4-(k-1)^4)*x^k/(1+x^k))/(1-x))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^4 - (d - 1)^4).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^4 - (k - 1)^4) * x^k/(1 + x^k).
a(n) ~ 7 * Pi^4 * n^4 / 720. - Vaclav Kotesovec, May 28 2021

A344723 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^5.

Original entry on oeis.org

1, 31, 243, 992, 3094, 7564, 16596, 31744, 58237, 97117, 158169, 241837, 364299, 521829, 745693, 1018120, 1389402, 1837302, 2423834, 3105432, 3998776, 5007286, 6289998, 7738784, 9543887, 11537207, 14031231, 16717879, 20018661, 23629281, 27958433, 32577739, 38219963, 44148743
Offset: 1

Views

Author

Seiichi Manyama, May 27 2021

Keywords

Comments

In general, for m > 1, Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^m ~ (1 - 2^(1-m)) * zeta(m) * n^m. - Vaclav Kotesovec, May 28 2021

Links

Crossrefs

Column k=5 of A344726.
Cf. A318744.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-3*DivisorSigma[0, n] + 2*DivisorSigma[0, 2*n] + 10*DivisorSigma[1, n] - 5*DivisorSigma[1, 2*n] - 15*DivisorSigma[2, n] + 5*DivisorSigma[2, 2*n] + 25/2 * DivisorSigma[3, n] - 5/2 * DivisorSigma[3, 2*n] - 45/8 *DivisorSigma[4, n] + 5/8 * DivisorSigma[4, 2*n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^5);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(d^5-(d-1)^5)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^5-(k-1)^5)*x^k/(1+x^k))/(1-x))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (d^5 - (d - 1)^5).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^5 - (k - 1)^5) * x^k/(1 + x^k).
a(n) ~ 15*zeta(5)*n^5/16. - Vaclav Kotesovec, May 28 2021
Showing 1-7 of 7 results.

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