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 1-6 of 6 results.

A108639 a(n) = Sum_{k=1..n} sigma_{n-k}(k), where sigma_m(k) = Sum_{j|k} j^m.

Original entry on oeis.org

1, 3, 6, 13, 29, 77, 229, 771, 2863, 11573, 50365, 234161, 1156039, 6031751, 33130187, 190929778, 1151198268, 7243777234, 47462906927, 323188163753, 2282922216819, 16701529748621, 126359471558613, 987316752551419
Offset: 1

Views

Author

Leroy Quet, Jul 06 2005

Keywords

Comments

Row sums of number triangle A109974. - Paul Barry, Jul 06 2005

Examples

			a(5) = 1^4 + (1^3 + 2^3) + (1^2 + 3^2) + (1^1 + 2^1 + 4^1) + (1^0 + 5^0) = 1 + 1 + 8 + 1 + 9 + 1 + 2 + 4 + 1 + 1 = 29.

Links

Crossrefs

Cf. A109974, A245466 (with k instead of n-k).

Programs

  • Magma
    A108639:= func< n | (&+[DivisorSigma(j, n-j): j in [0..n-1]]) >;
[A108639(n): n in [1..30]]; // G. C. Greubel, Oct 18 2023
  • Maple
    with(numtheory): s:=proc(n,k) local div: div:=divisors(n): sum(div[j]^k,j=1..tau(n)) end: a:=n->sum(s(i,n-i),i=1..n): seq(a(n),n=1..27); # Emeric Deutsch, Jul 13 2005
  • Mathematica
    Array[Sum[DivisorSigma[# - k, k], {k, #}] &, 24] (* Michael De Vlieger, Dec 23 2017 *)
  • PARI
    a(n) = sum(k=1, n, sigma(k, n-k)); \\ Michel Marcus, Dec 24 2017
  • SageMath
    def A108639(n): return sum(sigma(n-j, j) for j in range(n))
[A108639(n) for n in range(1,31)] # G. C. Greubel, Oct 18 2023

Extensions

More terms from Emeric Deutsch, Jul 13 2005
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A334874 a(n) = sigma(1) - tau(2) + sigma(3) - tau(4) + sigma(5) - tau(6) + ... - (up to n).

Original entry on oeis.org

1, -1, 3, 0, 6, 2, 10, 6, 19, 15, 27, 21, 35, 31, 55, 50, 68, 62, 82, 76, 108, 104, 128, 120, 151, 147, 187, 181, 211, 203, 235, 229, 277, 273, 321, 312, 350, 346, 402, 394, 436, 428, 472, 466, 544, 540, 588, 578, 635, 629, 701, 695, 749, 741, 813, 805, 885, 881, 941, 929
Offset: 1

Views

Author

Wesley Ivan Hurt, May 13 2020

Keywords

Examples

			a(1) = sigma(1) = 1;
a(2) = sigma(1) - tau(2) = 1 - 2 = -1;
a(3) = sigma(1) - tau(2) + sigma(3) = 1 - 2 + 4 = 3;
a(4) = sigma(1) - tau(2) + sigma(3) - tau(4) = 1 - 2 + 4 - 3 = 0;

Links

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A245466.

Programs

  • Maple
    f:= proc(n) if n::odd then numtheory:-sigma(n) else -numtheory:-tau(n) fi end proc:
ListTools:-PartialSums(map(f,[$1..100])); # Robert Israel, May 15 2020
  • Mathematica
    Table[Sum[(-1)^(k + 1)*DivisorSigma[Mod[k, 2], k], {k, n}], {n, 100}]
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*sigma(k, k % 2)); \\ Michel Marcus, May 14 2020

Formula

a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_[k mod 2](k), where sigma_[0](n) = tau(n), the number of divisors of n and sigma_[1](n) = sigma(n), the sum of the divisors of n.
a(p^k) - a(p^k-1) = (p^(k+1)-1)/(p-1), where p is an odd prime and k is a positive integer. - Wesley Ivan Hurt, May 15 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

A343781 a(n) = Sum_{k=1..floor(n/2)} sigma_k(n-k), where sigma_k(n) is the sum of the k-th powers of the divisors of n.

Original entry on oeis.org

0, 1, 3, 9, 17, 55, 111, 457, 943, 4962, 11148, 69526, 159402, 1161340, 2765874, 22829766, 55192956, 510771772, 1257880780, 12870681814, 32042113008, 359566186586, 904795505226, 11043196798176, 28002785395660, 369463867367567, 943392140873807, 13378621275148931
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 29 2021

Keywords

Examples

			a(5) = 17; a(5) = Sum_{i=1..2} sigma_k(5-k) = sigma_1(4) + sigma_2(3) = (1+2+4) + (1^2+3^2) = 7 + 10 = 17.

Crossrefs

Cf. A245466.

Programs

  • Mathematica
    Table[Sum[DivisorSigma[i, n - i], {i, Floor[n/2]}], {n, 30}]
  • PARI
    a(n) = sum(k=1, n\2, sigma(n-k, k)); \\ Michel Marcus, Apr 29 2021

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
Showing 1-6 of 6 results.

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

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