A013954 -id:A013954 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Aug 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..3000

Cf. A013954.

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

[0] cat [n*DivisorSigma(6,n): n in [1..25]]; // Vincenzo Librandi, Aug 02 2025

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]

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

A386786 a(n) = n^4*sigma_6(n).

Original entry on oeis.org

0, 1, 1040, 59130, 1065216, 9766250, 61495200, 282477650, 1090785280, 3491573931, 10156900000, 25937439242, 62986222080, 137858520410, 293776756000, 577478362500, 1116964192256, 2015993983970, 3631236888240, 6131066388122, 10403165760000, 16702903444500, 26974936811680

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A280022, A386783, A280025, A386784, A386785, A386787, A386788.

Cf. A013954, A386750, A386777, A386780.

[0] cat [n^4*DivisorSigma(6, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025

Table[n^4*DivisorSigma[6, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 1013*x^k + 47840*x^(2*k) + 455192*x^(3*k) + 1310354*x^(4*k) + 1310354*x^(5*k) + 455192*x^(6*k) + 47840*x^(7*k) + 1013*x^(8*k) + x^(9*k))/(1 - x^k)^11, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^4*x^k*(1 + 1013*x^k + 47840*x^(2*k) + 455192*x^(3*k) + 1310354*x^(4*k) + 1310354*x^(5*k) + 455192*x^(6*k) + 47840*x^(7*k) + 1013*x^(8*k) + x^(9*k))/(1 - x^k)^11.
a(n) = n^4*A013954(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-10). - R. J. Mathar, Aug 03 2025

A013964 a(n) = sigma_16(n), the sum of the 16th powers of the divisors of n.

Original entry on oeis.org

1, 65537, 43046722, 4295032833, 152587890626, 2821153019714, 33232930569602, 281479271743489, 1853020231898563, 10000152587956162, 45949729863572162, 184887084343023426, 665416609183179842, 2177986570740006274, 6568408508343827972, 18447025552981295105

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013675, A013954-A013972, A017665-A017712.

[DivisorSigma(16, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

DivisorSigma[16, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^16*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 16); \\ Amiram Eldar, Oct 29 2023

[sigma(n,16)for n in range(1,14)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^16*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-16)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(16*e+16)-1)/(p^16-1).
Sum_{k=1..n} a(k) = zeta(17) * n^17 / 17 + O(n^18). (End)

A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

Original entry on oeis.org

1, 1048577, 3486784402, 1099512676353, 95367431640626, 3656161927895954, 79792266297612002, 1152922604119523329, 12157665462543713203, 100000095367432689202, 672749994932560009202, 3833763649708914645906, 19004963774880799438802, 83668335217551100221154

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(20,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

DivisorSigma[20,Range[20]] (* Harvey P. Dale, Jul 26 2015 *)

vector(50, n, sigma(n,20)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,20)for n in range(1,13)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^20*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(20*e+20)-1)/(p^20-1).
Dirichlet g.f.: zeta(s)*zeta(s-20).
Sum_{k=1..n} a(k) = zeta(21) * n^21 / 21 + O(n^22). (End)

A013970 a(n) = sigma_22(n), the sum of the 22nd powers of the divisors of n.

Original entry on oeis.org

1, 4194305, 31381059610, 17592190238721, 2384185791015626, 131621735227521050, 3909821048582988050, 73786993887028445185, 984770902214992292491, 10000002384185795209930, 81402749386839761113322, 552061570551763831158810, 3211838877954855105157370

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

[DivisorSigma(22,n): n in [1..50]]; // G. C. Greubel, Nov 03 2018

lst={};Do[AppendTo[lst,DivisorSigma[22,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
a[ n_] := DivisorSigma[ 22, n]; (* Michael Somos, Dec 19 2016 *)

vector(50, n, sigma(n,22)) \\ G. C. Greubel, Nov 03 2018

[sigma(n,22)for n in range(1,12)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^22*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(22*e+22)-1)/(p^22-1).
Dirichlet g.f.: zeta(s)*zeta(s-22).
Sum_{k=1..n} a(k) = zeta(23) * n^23 / 23 + O(n^24). (End)

A386777 a(n) = n^2*sigma_6(n).

Original entry on oeis.org

0, 1, 260, 6570, 66576, 390650, 1708200, 5764850, 17043520, 43105851, 101569000, 214359002, 437404320, 815730890, 1498861000, 2566570500, 4363141376, 6975757730, 11207521260, 16983563402, 26007914400, 37875064500, 55733340520, 78310985810, 111975926400, 152597656875

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A282097, A386745, A282099, A386747, A282751, A282753, A386778.

Cf. A013954, A386750.

[0] cat [n^2*DivisorSigma(6, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Table[n^2*DivisorSigma[6, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^8*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^8*x^k*(1 + x^k)/(1 - x^k)^3.
a(n) = n^2*A013954(n).
Dirichlet g.f.: zeta(s-2)*zeta(s-8). - R. J. Mathar, Aug 03 2025

A386780 a(n) = n^3*sigma_6(n).

Original entry on oeis.org

0, 1, 520, 19710, 266304, 1953250, 10249200, 40353950, 136348160, 387952659, 1015690000, 2357949022, 5248851840, 10604501570, 20984054000, 38498557500, 69810262016, 118587881410, 201735382680, 322687704638, 520158288000, 795376354500, 1226133491440, 1801152673630

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A282211, A386746, A282213, A386748, A282781, A386781, A386782.

Cf. A013954, A386750, A386777.

[0] cat [n^3*DivisorSigma(6, n): n in [1..35]]; // Vincenzo Librandi, Aug 04 2025

Table[n^3*DivisorSigma[6, n], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[k^9*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k^9*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4.
a(n) = n^3*A013954(n).
Dirichlet g.f.: zeta(s-3)*zeta(s-9). - R. J. Mathar, Aug 03 2025

A017669 Numerator of sum of -3rd powers of divisors of n.

Original entry on oeis.org

1, 9, 28, 73, 126, 7, 344, 585, 757, 567, 1332, 511, 2198, 387, 392, 4681, 4914, 757, 6860, 4599, 1376, 2997, 12168, 455, 15751, 9891, 20440, 3139, 24390, 147, 29792, 37449, 4144, 22113, 6192, 55261, 50654, 15435, 61544, 7371, 68922, 172, 79508, 24309, 10598

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A017670 (denominator), A002117, A013662.

[Numerator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Numerator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, numerator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017670(n) = zeta(3) (A002117).
Dirichlet g.f. of a(n)/A017670(n): zeta(s)*zeta(s+3).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017670(k) = zeta(4) (A013662). (End)

A017670 Denominator of sum of -3rd powers of divisors of n.

Original entry on oeis.org

1, 8, 27, 64, 125, 6, 343, 512, 729, 500, 1331, 432, 2197, 343, 375, 4096, 4913, 648, 6859, 4000, 1323, 2662, 12167, 384, 15625, 8788, 19683, 2744, 24389, 125, 29791, 32768, 3993, 19652, 6125, 46656, 50653, 13718, 59319, 6400, 68921, 147, 79507, 21296, 10125

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017669.

[Denominator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Denominator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, denominator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Denominator of Sum_{d|n} 1/d^3.

Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

A017671 Numerator of sum of -4th powers of divisors of n.

Original entry on oeis.org

1, 17, 82, 273, 626, 697, 2402, 4369, 6643, 5321, 14642, 3731, 28562, 20417, 51332, 69905, 83522, 112931, 130322, 85449, 196964, 124457, 279842, 179129, 391251, 242777, 538084, 46839, 707282, 218161, 923522, 1118481, 1200644, 41761, 1503652, 604513, 1874162

Offset: 1

N. J. A. Sloane

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

			1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A017672 (denominator), A013662, A013663.

[Numerator(DivisorSigma(4,n)/n^4): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Table[Numerator[DivisorSigma[-4, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, numerator(sigma(n, 4)/n^4)) \\ G. C. Greubel, Nov 08 2018

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^4*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018
From Amiram Eldar, Apr 02 2024: (Start)
sup_{n>=1} a(n)/A017672(n) = zeta(4) (A013662).
Dirichlet g.f. of a(n)/A017672(n): zeta(s)*zeta(s+4).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017672(k) = zeta(5) (A013663). (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A386786 a(n) = n^4*sigma_6(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A013964 a(n) = sigma_16(n), the sum of the 16th powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A013970 a(n) = sigma_22(n), the sum of the 22nd powers of the divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A386777 a(n) = n^2*sigma_6(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A386780 a(n) = n^3*sigma_6(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica