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 1-9 of 9 results.

A320940 a(n) = Sum_{d|n} d*sigma_n(d).

Original entry on oeis.org

1, 11, 85, 1127, 15631, 287021, 5764809, 135007759, 3487020610, 100146496681, 3138428376733, 107032667155169, 3937376385699303, 155582338242604221, 6568408966322733475, 295154660699054931999, 14063084452067724991027, 708239400347943609329270
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2018

Keywords

Examples

			a(6) = 1*sigma_6(1)+2*sigma_6(2)+3*sigma_6(3)+6*sigma_6(6) = 1+2*65+3*730+6*47450 = 287021.

Links

Crossrefs

Cf. A000203, A001001, A027847, A027848, A319647, A321141.

Programs

  • Magma
    [&+[d*DivisorSigma(n,d):d in Divisors(n)]:n in [1..18]]; // Marius A. Burtea, Feb 15 2020
  • Maple
    with(numtheory): seq(coeff(series(n*(-log(mul((1-x^k)^sigma[n](k),k=1..n))),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
  • Mathematica
    Table[Sum[d DivisorSigma[n, d], {d, Divisors[n]}] , {n, 18}]
Table[n SeriesCoefficient[-Log[Product[(1 - x^k)^DivisorSigma[n, k], {k, 1, n}]], {x, 0, n}], {n, 18}]
  • PARI
    a(n) = sumdiv(n, d, d*sigma(d, n)); \\ Michel Marcus, Oct 28 2018
  • Python
    from sympy import divisor_sigma, divisors
def A320940(n):
    return sum(divisor_sigma(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

a(n) = n * [x^n] -log(Product_{k>=1} (1 - x^k)^sigma_n(k)).
a(n) = Sum_{d|n} d^(n+1)*sigma_1(n/d).
a(n) ~ n^(n+1). - Vaclav Kotesovec, Feb 16 2020

A344434 a(n) = Sum_{d|n} sigma_d(d), where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 6, 29, 279, 3127, 47484, 823545, 16843288, 387440202, 10009769782, 285311670613, 8918294591103, 302875106592255, 11112685049470800, 437893920912789563, 18447025552998138393, 827240261886336764179, 39346558271492566413252, 1978419655660313589123981
Offset: 1

Views

Author

Wesley Ivan Hurt, May 19 2021

Keywords

Comments

Inverse Möbius transform of sigma_n(n) (A023887). - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(6) = Sum_{d|6} sigma_d(d) = (1^1) + (1^2 + 2^2) + (1^3 + 3^3) + (1^6 + 2^6 + 3^6 + 6^6) = 47484.

Links

Crossrefs

Cf. A023887 (sigma_n(n)), A245466, A321141, A334874, A343781.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 20}]
  • PARI
    a(n) = sumdiv(n, d, sigma(d, d)); \\ Michel Marcus, May 19 2021
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k))) \\ Seiichi Manyama, Jul 25 2022

Formula

If p is prime, a(p) = Sum_{d|p} sigma_d(d) = sigma_1(1) + sigma_p(p) = 1^1 + (1^p + p^p) = p^p + 2.
G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 25 2022

A356046 a(n) = Sum_{k=1..n} sigma_n(k) * floor(n/k).

Original entry on oeis.org

1, 7, 40, 393, 4498, 68898, 1205205, 24830617, 574911611, 14936215765, 427782762142, 13426870089265, 457622727372932, 16842615801316402, 665489035541044561, 28102162770144986248, 1262904298391426474369, 60182778141796948356895
Offset: 1

Views

Author

Seiichi Manyama, Jul 24 2022

Keywords

Links

Crossrefs

Cf. A321141, A356045.

Programs

  • PARI
    a(n) = sum(k=1, n, sigma(k, n)*(n\k));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, d^n*numdiv(k/d)));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, sigma(d, n)));

Formula

a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} sigma_n(k) * x^k/(1 - x^k).
a(n) = Sum_{k=1..n} Sum_{d|k} d^n * tau(k/d).
a(n) = Sum_{k=1..n} Sum_{d|k} sigma_n(d).
a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Aug 07 2022

A322103 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 1, 6, 5, 6, 1, 10, 11, 11, 3, 1, 18, 29, 27, 7, 9, 1, 34, 83, 83, 27, 20, 3, 1, 66, 245, 291, 127, 66, 9, 10, 1, 130, 731, 1091, 627, 290, 51, 26, 6, 1, 258, 2189, 4227, 3127, 1494, 345, 112, 18, 9, 1, 514, 6563, 16643, 15627, 8330, 2403, 668, 102, 28, 3
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2018

Keywords

Examples

			Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  3,   4,   6,   10,    18,    34,  ...
  3,   5,  11,   29,    83,   245,  ...
  6,  11,  27,   83,   291,  1091,  ...
  3,   7,  27,  127,   627,  3127,  ...
  9,  20,  66,  290,  1494,  8330,  ...

Links

Crossrefs

Columns k=0..3 give A007425, A007429, A007433, A321140.
Cf. A109974, A321141 (diagonal), A356045.

Programs

  • Mathematica
    Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
  • PARI
    T(n,k)={sumdiv(n, d, d^k*numdiv(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Formula

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

A344480 a(n) = Sum_{d|n} d * sigma_d(d), where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 11, 85, 1103, 15631, 284795, 5764809, 134745175, 3486961642, 100097682141, 3138428376733, 107019534806039, 3937376385699303, 155577590686826319, 6568408813691811835, 295152408847835466855, 14063084452067724991027, 708238048886862707907062, 37589973457545958193355621, 2097154000001929438984022793
Offset: 1

Views

Author

Wesley Ivan Hurt, May 20 2021

Keywords

Comments

If p is prime, a(p) = Sum_{d|p} d * sigma_d(d) = 1*(1^1) + p*(1^p + p^p) = 1 + p + p^(p+1).
Inverse Möbius transform of n * sigma_n(n). - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(6) = Sum_{d|6} d * sigma_d(d) = 1*(1^1) + 2*(1^2 + 2^2) + 3*(1^3 + 3^3) + 6*(1^6 + 2^6 + 3^6 + 6^6) = 284795.

Crossrefs

Cf. A245466, A321141, A334874, A343781, A344434.

Programs

  • Mathematica
    Table[Sum[k*DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 30}]
  • PARI
    a(n) = sumdiv(n, d, d*sigma(d, d)); \\ Michel Marcus, May 21 2021

A344787 a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 7, 31, 287, 3131, 47527, 823551, 16843583, 387440266, 10009772937, 285311670623, 8918294639219, 302875106592267, 11112685050294387, 437893920912795941, 18447025553014982271, 827240261886336764195, 39346558271492953948522, 1978419655660313589123999
Offset: 1

Views

Author

Wesley Ivan Hurt, May 28 2021

Keywords

Comments

If p is prime, a(p) = p * Sum_{d|p} sigma_d(d) / d = p * (1 + (1^p + p^p)/p) = 1 + p + p^p.

Examples

			a(4) = 4 * Sum_{d|4} sigma_d(d) / d = 4 * ((1^1)/1 + (1^2 + 2^2)/2 + (1^4 + 2^4 + 4^4)/4) = 287.

Links

Crossrefs

Cf. A245466, A321141, A334874, A343781, A344434, A344480.
Cf. A001001, A007429, A027847, A027848, A060640.

Programs

  • Mathematica
    Table[n*Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k])/k, {k, n}], {n, 20}]
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Dec 16 2022

Formula

G.f.: Sum_{k>=1} sigma_k(k) * x^k/(1 - x^k)^2. - Seiichi Manyama, Dec 16 2022

A348397 a(n) = Sum_{d|n} sigma_[n-d](d), where sigma_[k](n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 3, 3, 9, 3, 50, 3, 343, 734, 3388, 3, 133959, 3, 827646, 10297073, 33640713, 3, 2579172499, 3, 44822639761, 678610493345, 285312719194, 3, 393067887861756, 95367431640630, 302875123369476, 150094918113956098, 569940024192528003, 3, 105474401758856279784, 3
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 16 2021

Keywords

Examples

			a(6) = 50; a(6) = sigma_[6-1](1) + sigma_[6-2](2) + sigma_[6-3](3) + sigma_[6-6](6) = (1^5) + (1^4 + 2^4) + (1^3 + 3^3) + (6^0 + 6^0 + 6^0 + 6^0) = 50.

Links

Crossrefs

Cf. A321141.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[n - #, #] &]; Array[a, 30] (* Amiram Eldar, Oct 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d, n-d)); \\ Michel Marcus, Oct 18 2021

Formula

a(n) = 3 iff n is prime. - Bernard Schott, Oct 17 2021

A348399 a(n) = Sum_{d|n} sigma_[d](n), where sigma_[k](n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 8, 32, 301, 3132, 47764, 823552, 16847478, 387440943, 10009869956, 285311670624, 8918297605544, 302875106592268, 11112685154884700, 437893920913552704, 18447025557293175687, 827240261886336764196, 39346558271690970332766, 1978419655660313589124000
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 16 2021

Keywords

Examples

			a(4) = 301; a(4) = sigma_[1](4) + sigma_[2](4) + sigma_[4](4) = (1^1 + 2^1 + 4^1) + (1^2 + 2^2 + 4^2) + (1^4 + 2^4 + 4^4) = 301.

Crossrefs

Cf. A321141.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[#, n] &]; Array[a, 20] (* Amiram Eldar, Oct 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(n, d)); \\ Michel Marcus, Oct 18 2021

Formula

a(p) = p^p + p + 2 for primes p, since we have a(p) = sigma_[1](p) + sigma[p](p) = (1 + p) + (1^p + p^p) = p^p + p + 2. - Wesley Ivan Hurt, Nov 03 2021

A348398 a(n) = Sum_{d|n} sigma_[n/d](d), where sigma_[k](n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

1, 4, 5, 13, 7, 32, 9, 54, 42, 78, 13, 299, 15, 204, 395, 647, 19, 1626, 21, 2881, 2565, 2208, 25, 17070, 3158, 8406, 20482, 35607, 31, 116964, 33, 136104, 178529, 131418, 94983, 1112928, 39, 524712, 1596579, 2533908, 43, 7283718, 45, 8405995, 16364934, 8389212, 49, 78586033, 823602, 43423962
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 16 2021

Keywords

Examples

			a(8) = 54; a(8) = sigma_[8/1](1) + sigma_[8/2](2) + sigma_[8/4](4) + sigma_[8/8](8) = (1^8) + (1^4 + 2^4) + (1^2 + 2^2 + 4^2) + (1^1 + 2^1 + 4^1 + 8^1) = 54.

Crossrefs

Cf. A321141.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, DivisorSigma[n/#, #] &]; Array[a, 50] (* Amiram Eldar, Oct 17 2021 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d, n/d)); \\ Michel Marcus, Oct 18 2021
Showing 1-9 of 9 results.

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