A267316 -id:A267316 - cp's OEIS frontend

A000584 Fifth powers: a(n) = n^5.

0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, 11881376, 14348907, 17210368, 20511149

Offset: 0

N. J. A. Sloane

Totally multiplicative sequence with a(p) = p^5 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A059338. The inverse binomial transform yields the (finite) 0, 1, 30, 150, 240, 120, the 5th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Equals sum of odd numbers from n^2*(n-1)+1 (A100104) to n^2*(n+1)-1 (A003777). - Bruno Berselli, Mar 14 2014
a(n) mod 10 = n mod 10. - Reinhard Zumkeller, May 10 2014
Numbers of the form a(n) + a(n+1) + ... + a(n+k) are nonprime for all n, k>=0; this can be proved by the method indicated in the comment in A256581. - Vladimir Shevelev and Peter J. C. Moses, Apr 04 2015

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..500
Brady Haran and Simon Pampena, Fifth Root Trick, Numberphile video (2014)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Partial sums give A000539.
Cf. A000012, A001477, A000290, A000578, A000583, A062392, A022521 (first differences), A008292, A173018, A123125.
Cf. A013663, A162624, A267316.

a000584 = (^ 5)  -- Reinhard Zumkeller, Nov 11 2012

Range[0,50]^5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

makelist(n^5, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

a(n)=n^5 \\ Charles R Greathouse IV, Jul 28 2015

[n**5 for n in range(30)]  # Zerinvary Lajos, Jun 03 2009

G.f.: x*(1+26*x+66*x^2+26*x^3+x^4) / (x-1)^6. [Simon Plouffe in his 1992 dissertation]
Multiplicative with a(p^e) = p^(5e). - David W. Wilson, Aug 01 2001
E.g.f.: exp(x)*(x+15*x^2+25*x^3+10*x^4+x^5). - Geoffrey Critzer, Jun 12 2013
a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 120. - Ant King, Sep 23 2013
a(n) = n + Sum_{j=0..n-1}{k=1..4}binomial(5,k)*j^(5-k). - Patrick J. McNab, Mar 28 2016
From Kolosov Petro, Oct 22 2018: (Start)
a(n) = Sum_{k=1..n} A300656(n,k).
a(n) = Sum_{k=0..n-1} A300656(n,k). (End)
a(n) = Sum_{k=1..5} Eulerian(5, k)*binomial(n+5-k, 5), with Eulerian(5, k) = A008292(5, k), the numbers 1, 26, 66, 26, 1, for n >= 0. Worpitzki's identity for powers of 5. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - Wolfdieter Lang, Jul 17 2019
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(5) (A013663).
Sum_{n>=1} (-1)^(n+1)/a(n) = 15*zeta(5)/16 (A267316). (End)

More terms from Henry Bottomley, Jun 21 2001

A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

Original entry on oeis.org

1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136685.

Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = a(n)/A334604(n). - _Petros Hadjicostas_, May 07 2020

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k,1,n} ] ], {n,1,30} ]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020

A275703 Decimal expansion of the Dirichlet eta function at 6.

Original entry on oeis.org

9, 8, 5, 5, 5, 1, 0, 9, 1, 2, 9, 7, 4, 3, 5, 1, 0, 4, 0, 9, 8, 4, 3, 9, 2, 4, 4, 4, 8, 4, 9, 5, 4, 2, 6, 1, 4, 0, 4, 8, 8, 5, 6, 9, 3, 4, 6, 9, 3, 2, 6, 8, 8, 8, 0, 3, 4, 8, 3, 3, 3, 9, 3, 2, 5, 4, 1, 9, 6, 8, 0, 2, 1, 8, 6, 2, 7, 1, 7, 1, 3, 5, 7, 3, 9, 3, 7, 2, 9, 1, 1, 2, 7, 9, 5, 5, 9, 4, 6, 4

Offset: 0

Terry D. Grant, Aug 05 2016

It appears that each sum of a Dirichlet eta function is 1/2^(x-1) less than the zeta(x), where x is a positive integer > 1. In this case, eta(x) = eta(6) = (31/32)*zeta(6) = 31*(Pi^6)/30240. Therefore eta(6) = 1/2^(6-1) or 1/32nd less than zeta(6) (see A013664). [Edited by Petros Hadjicostas, May 07 2020]

			31*(Pi^6)/30240 = 0.9855510912974...

Cf. A002162 (decimal expansion of value at 1), A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275710 (value at 7).

Cf. A013664, A136677, A334605.

Mathematica
```
RealDigits[31*(Pi^6)/30240,10,100]
```

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
print(s) # Terry D. Grant, Aug 05 2016

eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.

eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - Petros Hadjicostas, May 07 2020

A275710 Decimal expansion of the Dirichlet eta function at 7.

Original entry on oeis.org

9, 9, 2, 5, 9, 3, 8, 1, 9, 9, 2, 2, 8, 3, 0, 2, 8, 2, 6, 7, 0, 4, 2, 5, 7, 1, 3, 1, 3, 3, 3, 9, 3, 6, 8, 5, 2, 3, 1, 1, 1, 5, 6, 9, 2, 4, 3, 1, 4, 0, 6, 8, 5, 1, 6, 2, 9, 5, 1, 3, 0, 8, 7, 5, 6, 2, 6, 7, 0, 2, 0, 5, 2, 1, 8, 6, 4, 7, 0, 5, 1, 9, 8, 1, 3, 1, 4, 2, 0, 3, 7, 7, 4, 5, 7, 2, 3, 9, 7, 0

Offset: 0

Terry D. Grant, Aug 06 2016

			0.99259381992283028267...

Cf. A002162 (value at 1), A013665, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275703 (value at 6), A334668, A334669, A347150, A347059.

RealDigits[63 Zeta[7]/64, 10, 100] [[1]]

-polylog(7, -1) \\ Michel Marcus, Aug 20 2021

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^7
print(s) # Terry D. Grant, Aug 06 2016

eta(7) = 63*zeta(7)/64 = (63*A013665)/64.
eta(7) = Lim_{n -> infinity} A334668(n)/A334669(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{k>=1} (-1)^(k+1) / k^7. - Sean A. Irvine, Aug 19 2021

A334604 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^5.

Original entry on oeis.org

1, 32, 7776, 248832, 777600000, 777600000, 13069123200000, 418211942400000, 101625502003200000, 101625502003200000, 16366888723117363200000, 16366888723117363200000, 6076911214672415134617600000

Offset: 1

Petros Hadjicostas, May 07 2020

nonn

Lim_{n -> infinity} A136676(n)/a(n) = A267316 = (15/16)*A013663.

			The first few fractions are 1, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = A136676/A334604.

Cf. A013663, A136676 (numerators), A267316.

Denominator @ Accumulate[Table[(-1)^(k + 1)/k^5, {k, 1, 13}]] (* Amiram Eldar, May 08 2020 *)

a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020

A347059 Decimal expansion of the Dirichlet eta function at 9.

Original entry on oeis.org

9, 9, 8, 0, 9, 4, 2, 9, 7, 5, 4, 1, 6, 0, 5, 3, 3, 0, 7, 6, 7, 7, 8, 3, 0, 3, 1, 8, 5, 2, 5, 9, 7, 9, 5, 0, 8, 7, 4, 3, 3, 3, 9, 5, 3, 5, 3, 7, 8, 7, 7, 4, 7, 2, 3, 4, 3, 3, 2, 8, 6, 6, 0, 3, 7, 8, 8, 8, 7, 4, 5, 5, 5, 2, 5, 4, 5, 2, 7, 0, 2, 0, 7, 9, 4, 9, 3

Offset: 0

Sean A. Irvine, Aug 14 2021

			0.998094297541605330767783031852597950...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A347150, A346927.

First[RealDigits[N[DirichletEta[9],87]]] (* Stefano Spezia, Aug 15 2021 *)

-polylog(9, -1) \\ Michel Marcus, Aug 15 2021

Equals (255/256) * zeta(9).
Equals Sum_{k>=1} (-1)^(k+1) / k^9.
Equals eta(9).

A347150 Decimal expansion of the Dirichlet eta function at 8.

Original entry on oeis.org

9, 9, 6, 2, 3, 3, 0, 0, 1, 8, 5, 2, 6, 4, 7, 8, 9, 9, 2, 2, 7, 2, 8, 9, 2, 6, 0, 0, 8, 2, 8, 0, 3, 6, 1, 7, 8, 7, 4, 1, 2, 5, 1, 5, 9, 4, 7, 2, 8, 9, 8, 0, 6, 7, 0, 4, 5, 2, 8, 9, 0, 2, 9, 1, 9, 4, 3, 5, 9, 6, 4, 8, 2, 5, 7, 7, 5, 8, 5, 8, 9, 2, 8, 2, 8, 2, 4

Offset: 0

Sean A. Irvine, Aug 19 2021

			0.9962330018526478992272892600828036178741251594728980...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A346927, A347059.

RealDigits[DirichletEta[8], 10, 100][[1]] (* Amiram Eldar, Aug 20 2021 *)

-polylog(8, -1) \\ Michel Marcus, Aug 20 2021

Equals (127/128) * zeta(8).
Equals 127 * Pi^8 / 1209600.
Equals Sum_{k>=1} (-1)^(k+1) / k^8.
Equals eta(8).

A346927 Decimal expansion of the Dirichlet eta function at 10.

Original entry on oeis.org

9, 9, 9, 0, 3, 9, 5, 0, 7, 5, 9, 8, 2, 7, 1, 5, 6, 5, 6, 3, 9, 2, 2, 1, 8, 4, 5, 6, 9, 9, 3, 4, 1, 8, 3, 1, 4, 2, 5, 9, 2, 9, 6, 4, 9, 6, 6, 6, 8, 9, 0, 6, 4, 7, 1, 0, 6, 8, 9, 4, 8, 7, 5, 5, 0, 6, 1, 4, 2, 4, 5, 8, 3, 8, 4, 0, 3, 8, 1, 2, 4, 4, 0, 7, 9, 8, 5

Offset: 0

Sean A. Irvine, Aug 07 2021

			0.999039507598271565639221845699341831425929649666890...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (306).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A072691, A197070, A267315, A267316, A275703, A275710, A347150, A347059.

RealDigits[DirichletEta[10], 10, 100][[1]] (* Amiram Eldar, Aug 08 2021 *)

-polylog(10, -1) \\ Michel Marcus, Aug 08 2021

Equals 73 * Pi^10 / (2^9 * 3^5 * 5 * 11).
Equals (511/512) * zeta(10).
Equals Sum_{k>=1} (-1)^(k+1) / k^10.
Equals eta(10).

A246967 Decimal expansion of the real positive solution to eta(x) = x.

Original entry on oeis.org

6, 2, 9, 3, 3, 4, 0, 9, 4, 0, 0, 9, 3, 7, 2, 7, 6, 7, 5, 5, 6, 4, 8, 0, 5, 0, 2, 5, 8, 9, 3, 2, 6, 1, 3, 7, 6, 4, 7, 2, 0, 7, 6, 4, 6, 8, 6, 6, 1, 8, 5, 3, 5, 5, 0, 6, 8, 8, 5, 8, 0, 2, 3, 1, 9, 7, 2, 6, 9, 2, 8, 5, 2, 9, 1, 5, 5, 7, 4, 6, 2, 1, 1, 0, 4, 2, 0, 0, 7, 9, 7, 5, 5, 6, 1, 9, 4

Offset: 0

Michal Paulovic, Sep 08 2014

Fixed point of Dirichlet eta function.

			0.6293340940...

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.
Wikipedia, Dirichlet Eta Function.

Cf. A197070, A267315, A267316.

RealDigits[x /. FindRoot[DirichletEta[x] - x, {x, 0}, WorkingPrecision -> 120], 10, 100] [[1]] (* Amiram Eldar, May 24 2021 *)

solve(n=0,2,(1-2^(1-n))*zeta(n)-n) \\ Edward Jiang, Sep 08 2014

A269482 Continued fraction expansion of the Dirichlet eta function at 5.

Original entry on oeis.org

0, 1, 34, 1, 6, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 35, 3, 1, 5, 4, 1, 2, 2, 1, 4, 1, 1, 1, 2, 10, 2, 1, 6, 9, 23, 1, 5, 1, 1, 1, 1, 1, 2, 1, 3, 4, 1, 2, 1, 1, 2, 2, 1, 1, 5, 4, 7, 1, 1, 1, 1, 2, 2, 1, 4, 1, 1, 2, 8, 3, 2, 1, 3, 1, 5, 356, 2, 57, 6, 1, 6, 1, 1, 31, 1, 5, 1, 1, 477, 1, 9, 7, 3, 4

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

Continued fraction of Sum_{k>=1} (-1)^(k - 1)/k^5 = (15*zeta(5))/16 = 0.9721197704469093...

			1/1^5 - 1/2^5 + 1/3^5 - 1/4^5 + 1/5^5 - 1/6^5 +... = 1/(1 + 1/(34 + 1/(1 + 1/(6 + 1/(1 + 1/(1 + 1/...)))))).

OEIS Wiki, Euler's alternating zeta function
Eric Weisstein's World of Mathematics, Dirichlet Eta Function
Wikipedia, Dirichlet Eta Function
Index entries for continued fractions for constants

Cf. A013681, A267316.

Mathematica
```
ContinuedFraction[(15 Zeta[5])/16, 100]
```

cp's OEIS Frontend

A000584 Fifth powers: a(n) = n^5.

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A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.

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A275703 Decimal expansion of the Dirichlet eta function at 6.

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A275710 Decimal expansion of the Dirichlet eta function at 7.

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A334604 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^5.

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A347059 Decimal expansion of the Dirichlet eta function at 9.

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A347150 Decimal expansion of the Dirichlet eta function at 8.

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