A002594 -id:A002594 - cp's OEIS frontend

A081864 Sum of 5th powers of the divisors of odd numbers: a(n) = sigma_5(2n-1).

1, 244, 3126, 16808, 59293, 161052, 371294, 762744, 1419858, 2476100, 4101152, 6436344, 9768751, 14408200, 20511150, 28629152, 39296688, 52541808, 69343958, 90595736, 115856202, 147008444, 185349918, 229345008, 282492057, 346445352, 418195494, 503448552, 604168400

Offset: 1

N. J. A. Sloane. This sequence was in the 1973 "Handbook", but then was deleted from the database because of confusion with A002594. Resubmitted by Benoit Cloitre, Apr 12 2003. Entry revised by N. J. A. Sloane, Jun 10 2012

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 56).
Index entries for sequences mentioned by Glaisher.

Cf. A001160, A333972.

[DivisorSigma(5,2*n-1):n in [1..30]]; // Marius A. Burtea, Aug 17 2019

with(numtheory); [seq(sigma[5](2*n-1),n=1..100)];

DivisorSigma[5,Range[1,61,2]] (* Harvey P. Dale, Jan 12 2016 *)

a(n) = sigma(2*n-1, 5); \\ Michel Marcus, Aug 17 2019

Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^6 / 180 = 5.341051... = 3*A333972. - Amiram Eldar, Jan 08 2025

Definition clarified by Harvey P. Dale, Jan 12 2016

A265021 Sum of fifth powers of the first n even numbers.

Original entry on oeis.org

0, 32, 1056, 8832, 41600, 141600, 390432, 928256, 1976832, 3866400, 7066400, 12220032, 20182656, 32064032, 49274400, 73574400, 107128832, 152564256, 213030432, 292265600, 394665600, 525356832, 690273056, 896236032, 1151040000, 1463540000, 1843744032

Offset: 0

Assoul Abdelkarim, Nov 30 2015

nonn

			a(4) =  2^5 + 4^5 + 6^5 + 8^5 = 41600.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Assoul Abdelkarim, The sums of powers of integers natural even, odd, RMSS-D-15-00105.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000539, A002594 (the same for odd numbers).

[(8/3)*n^2*(n+1)^2*(2*n^2+2*n-1): n in [0..30]]; // Vincenzo Librandi, Dec 01 2015

Accumulate[Range[0, 60, 2]^5] (* Michael De Vlieger, Nov 30 2015 *)
CoefficientList[Series[32 x (1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^7, {x, 0, 33}], x] (* Vincenzo Librandi, Dec 01 2015 *)

vector(100, n, n--; (8/3)*n^2*(n+1)^2*(2*n^2+2*n-1)) \\ Altug Alkan, Dec 01 2015

a(n) = 32 * Sum_{i=0..n} i^5 = (8/3)*n^2*(n+1)^2*(2*n^2+2*n-1).
a(n) = 32 * A000539(n).
G.f.: 32*x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^7. - Vincenzo Librandi, Dec 01 2015
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Vincenzo Librandi, Dec 01 2015

A349490 Sum of the n-th powers of the first n odd numbers.

Original entry on oeis.org

1, 10, 153, 3108, 79225, 2437006, 87922513, 3642188936, 170423429841, 8891285549650, 511800291063721, 32222868466588460, 2202868653419747209, 162509566498902542934, 12868118600594014094625, 1088626666242258867384848, 97994054039441765759931169

Offset: 1

Seiichi Manyama, Dec 09 2021

nonn

Sum of the k-th powers of the first n odd numbers: A000290 (k=1), A000447 (k=2), A002593 (k=3), A002309 (k=4), A002594 (k=5), A259322 (k=6).

Cf. A005408, A031971, A061787, A103438.

Table[Sum[(2*k-1)^n, {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Dec 09 2021 *)

PARI
```
a(n) = sum(k=1, n, (2*k-1)^n);
```

a(n) = Sum_{k=1..n} (2*k-1)^n.

a(n) ~ 2^n * n^n / (exp(1/2) - exp(-1/2)). - Vaclav Kotesovec, Dec 09 2021

cp's OEIS Frontend

A081864 Sum of 5th powers of the divisors of odd numbers: a(n) = sigma_5(2n-1).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A265021 Sum of fifth powers of the first n even numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A349490 Sum of the n-th powers of the first n odd numbers.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula