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 1-10 of 10 results.

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400
Offset: 0

Views

Author

N. J. A. Sloane, Feb 04 2017

Keywords

Comments

The q-expansion of the square of this expression is given in A281371.
Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

Links

Crossrefs

Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.

Programs

  • Magma
    [0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018
  • Maple
    with(gfun):
with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)/720,q,M+1);
seriestolist(t1);
# alternative program
seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024
  • Mathematica
    Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *)
terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
  • PARI
    for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017

Formula

a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017
G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017
a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017
G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

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

A386746 a(n) = n^3*sigma_2(n).

Original entry on oeis.org

0, 1, 40, 270, 1344, 3250, 10800, 17150, 43520, 66339, 130000, 162382, 362880, 373490, 686000, 877500, 1396736, 1424770, 2653560, 2482958, 4368000, 4630500, 6495280, 6448510, 11750400, 10171875, 14939600, 16140060, 23049600, 20535538, 35100000, 28658942, 44728320
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A001157, A328259, A386745.

Programs

  • Magma
    [0] cat [n^3*DivisorSigma(2, n): n in [1..35]]; // Vincenzo Librandi, Aug 02 2025
  • Mathematica
    Table[n^3*DivisorSigma[2, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^5*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^5*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Amiram Eldar, Aug 01 2025
a(n) = n^3*A001157(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-5). - R. J. Mathar, Aug 03 2025

A386749 a(n) = n*sigma_4(n).

Original entry on oeis.org

0, 1, 34, 246, 1092, 3130, 8364, 16814, 34952, 59787, 106420, 161062, 268632, 371306, 571676, 769980, 1118480, 1419874, 2032758, 2476118, 3417960, 4136244, 5476108, 6436366, 8598192, 9781275, 12624404, 14528268, 18360888, 20511178, 26179320, 28629182, 35791392, 39621252
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A064987, A328259, A281372, A282050, A386750, A282060, A386751.
Cf. A001159, A386747, A386748.

Programs

  • Magma
    [0] cat [n*DivisorSigma(4, n): n in [1..35]]; // Vincenzo Librandi, Aug 02 2025
  • Mathematica
    Table[n*DivisorSigma[4, n], {n, 0, 50}]
nmax = 50; CoefficientList[Series[x*Sum[k^5*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    a(n) = if (n, n*sigma(n,4), 0); \\ Michel Marcus, Aug 02 2025

Formula

G.f.: Sum_{k>=1} k^5*x^(k-1)/(1 - x^k)^2.
a(n) = n*A001159(n).
Dirichlet g.f.: zeta(s-1)*zeta(s-5). - R. J. Mathar, Aug 03 2025

A386750 a(n) = n*sigma_6(n).

Original entry on oeis.org

0, 1, 130, 2190, 16644, 78130, 284700, 823550, 2130440, 4789539, 10156900, 19487182, 36450360, 62748530, 107061500, 171104700, 272696336, 410338690, 622640070, 893871758, 1300395720, 1803574500, 2533333660, 3404825470, 4665663600, 6103906275, 8157308900, 10474721820
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A013954.
Cf. A064987, A328259, A281372, A386749, A282050, A282060, A386751.

Programs

  • Magma
    [0] cat [n*DivisorSigma(6,n): n in [1..25]]; // Vincenzo Librandi, Aug 02 2025
  • Mathematica
    Table[n*DivisorSigma[6, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[x*Sum[k^7*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^7*x^(k-1)/(1 - x^k)^2.
a(n) = n*A013954(n).
Dirichlet g.f.: zeta(s-1)*zeta(s-7). - R. J. Mathar, Aug 03 2025

A386751 a(n) = n*sigma_8(n).

Original entry on oeis.org

0, 1, 514, 19686, 263172, 1953130, 10118604, 40353614, 134744072, 387479547, 1003908820, 2357947702, 5180803992, 10604499386, 20741757596, 38449317180, 68988964880, 118587876514, 199164487158, 322687697798, 514009128360, 794401245204, 1211985118828, 1801152661486
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A013956.
Cf. A064987, A328259, A281372, A386749, A282050, A386750, A282060.

Programs

  • Magma
    [0] cat [n*DivisorSigma(8,n): n in [1..25]]; // Vincenzo Librandi, Aug 02 2025
  • Mathematica
    Table[n*DivisorSigma[8, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[x*Sum[k^9*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^9*x^(k-1)/(1 - x^k)^2.
a(n) = n*A013956(n).
Dirichlet g.f.: zeta(s-1)*zeta(s-9). - R. J. Mathar, Aug 03 2025

A386783 a(n) = n^4*sigma_2(n).

Original entry on oeis.org

0, 1, 80, 810, 5376, 16250, 64800, 120050, 348160, 597051, 1300000, 1786202, 4354560, 4855370, 9604000, 13162500, 22347776, 24221090, 47764080, 47176202, 87360000, 97240500, 142896160, 148315730, 282009600, 254296875, 388429600, 435781620, 645388800, 595530602, 1053000000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A280022, A280025, A386784, A386785, A386786, A386787, A386788.
Cf. A001157, A328259, A386745, A386746.

Programs

  • Magma
    [0] cat [n^4*DivisorSigma(2, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025
  • Mathematica
    Table[n^4*DivisorSigma[2, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^4*x^k*(1 + 57*x^k + 302*x^(2*k) + 302*x^(3*k) + 57*x^(4*k) + x^(5*k)) / (1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + 57*x^k + 302*x^(2*k) + 302*x^(3*k) + 57*x^(4*k) + x^(5*k)) / (1 - x^k)^7.
a(n) = n^4*A001157(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-6). - R. J. Mathar, Aug 03 2025

A386745 a(n) = n^2*sigma_2(n).

Original entry on oeis.org

0, 1, 20, 90, 336, 650, 1800, 2450, 5440, 7371, 13000, 14762, 30240, 28730, 49000, 58500, 87296, 83810, 147420, 130682, 218400, 220500, 295240, 280370, 489600, 406875, 574600, 597780, 823200, 708122, 1170000, 924482, 1397760, 1328580, 1676200, 1592500, 2476656
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A001157, A328259, A386746.

Programs

  • Magma
    [0] cat [n^2*DivisorSigma(2, n): n in [1..45]]; // Vincenzo Librandi, Aug 04 2025
  • Mathematica
    Table[n^2*DivisorSigma[2, n], {n, 0, 40}]
nmax = 40; CoefficientList[Series[Sum[k^4*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    a(n) = if (n, n^2*sigma(n, 2), 0); \\ Michel Marcus, Aug 01 2025

Formula

G.f.: Sum_{k>=1} k^4*x^k*(1 + x^k)/(1 - x^k)^3. - Amiram Eldar, Aug 01 2025
a(n) = n^2*A001157(n).
Dirichlet g.f.: zeta(s-2)*zeta(s-4). - R. J. Mathar, Aug 03 2025

A366135 Expansion of Sum_{k>=1} k^3 * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 11, 33, 98, 140, 366, 371, 820, 936, 1550, 1397, 3276, 2288, 4102, 4650, 6696, 5066, 10413, 7049, 13860, 12306, 15422, 12443, 27480, 17825, 25246, 25650, 36652, 24824, 51900, 30287, 54096, 46266, 55862, 52150, 93366, 51356, 77710, 75738, 116200, 69782, 137172
Offset: 1

Views

Author

Seiichi Manyama, Oct 28 2023

Keywords

Links

Crossrefs

Cf. A007437, A309731, A309732.
Cf. A001158, A328259.
Cf. A000203, A001157.

Programs

  • Mathematica
    a[n_] := (n * DivisorSigma[1, n] + DivisorSigma[2, n]) * n/2; Array[a, 50] (* Amiram Eldar, Dec 15 2023 *)
  • PARI
    a(n) = n*(n*sigma(n)+sigma(n, 2))/2;

Formula

a(n) = n * (n * sigma(n) + sigma_2(n))/2.
a(n) = Sum_{d|n} d^3 * binomial(n/d+1,2).
a(n) = Sum_{k=1..n} k*sigma_2(gcd(n,k)).
Sum_{k=1..n} a(k) ~ (Pi^2/48 + zeta(3)/8) * n^4. - Amiram Eldar, Dec 15 2023

A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).

Original entry on oeis.org

1, 22, 105, 414, 745, 2310, 2737, 7134, 9231, 16390, 15961, 43470, 30745, 60214, 78225, 118238, 88417, 203082, 137161, 308430, 287385, 351142, 291985, 749070, 481245, 676390, 767391, 1133118, 731641, 1720950, 953281, 1924574, 1675905, 1945174, 2039065, 3821634
Offset: 1

Views

Author

Seiichi Manyama, May 26 2024

Keywords

Links

Crossrefs

Cf. A059376, A328259, A372952.
Cf. A373131, A373133.
Cf. A013661, A013663.

Programs

  • Mathematica
    f[p_, e_] := (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)
  • PARI
    J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=1) = sumdiv(n, d, J(d, k)*sigma(d^m));

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n) )^3 ).
a(n) = Sum_{d|n} J_3(d) * sigma(d), where the Jordan totient function J_3(n) = A059376(n)
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 1/p^4 - 1/p^5 + 1/p^6) = 1.54488120152452251241... . (End)
Showing 1-10 of 10 results.

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

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