A275703 -id:A275703 - cp's OEIS frontend

A001014 Sixth powers: a(n) = n^6.

0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776, 387420489, 481890304

Offset: 0

N. J. A. Sloane

Numbers both square and cubic. - Patrick De Geest
Totally multiplicative sequence with a(p) = p^6 for prime p. - Jaroslav Krizek, Nov 01 2009
Numbers n for which the order of the torsion subgroup of the elliptic curve y^2 = x^3 + n is t = 6, cf. Gebel link. - Artur Jasinski, Jun 30 2010
Note that Sum_{n>=1} 1/a(n) = Pi^6 / 945. - Mohammad K. Azarian, Nov 01 2011
The binomial transform yields A056468. The inverse binomial transform yields the (finite) 0, 1, 62, 540, ..., 720, the 6th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
For n > 0, a(n) is the largest number k such that k + n^3 divides k^2 + n^3. - Derek Orr, Oct 01 2014

			The 6th powers of the first few integers are: 0^6 = 0 = a(0), 1^6 = 1 = a(1), 2^6 = 64 = a(2), 3^6 = 9^3 = 729 = a(3), 4^6 = 2^12 = 4096 = a(4), 5^6 = 25^3 = 15625 = a(5), etc.

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, eq. (6.37).
Granino A. Korn and Theresa M.Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
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
Henry Bottomley, Illustration of initial terms
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Richard J. Mathar, Construction of Bhaskara pairs, arXiv:1703.01677 [math.NT], 2017.
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 (7, -21, 35, -35, 21, -7, 1).

Subsequence of A201217.
Cf. A000540 (partial sums), A022522 (first differences), A008292.
Intersection of A000290 (squares) and A000578 (cubes).
Cf. A002604 (n^6+1), A123866 (n^6-1), A013664 (zeta(6)), A275703 (eta(6)).
Cf. A003358 - A003368 (sums of 2, ..., 12 positive sixth powers).

a001014 n = a001014_list !! n
a001014_list = map (^ 6) [0..] -- Reinhard Zumkeller, Dec 04 2011

Table[n^6, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

A001014(n):=n^6$
makelist(A001014(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A001014(n)=n^6 \\ Charles R Greathouse IV, Sep 24 2015

A001014 = lambda n: n**6 # M. F. Hasler, Jul 03 2025

a(n) = A123866(n) + 1 = A002604(n) - 1.
G.f.: -x*(1+x)*(x^4+56*x^3+246*x^2+56*x+1) / (x-1)^7. - Simon Plouffe in his 1992 dissertation
Multiplicative with a(p^e) = p^(6e). - David W. Wilson, Aug 01 2001
E.g.f.: (x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6)*exp(x).  Generally, the e.g.f. for n^m is Sum_{k=1..m} A008277(m,k)*x^k*exp(x). - Geoffrey Critzer, Aug 25 2013
From Ant King, Sep 23 2013: (Start)
Signature {7, -21, 35, -35, 21, -7, 1}.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + 720. (End)
a(n) == 1 (mod 7) if gcd(n, 7) = 1, otherwise a(n) == 0 (mod 7). See A109720. - Jake Lawrence, May 28 2016
From Ilya Gutkovskiy, Jul 06 2016: (Start)
Dirichlet g.f.: zeta(s-6).
Sum_{n>=1} 1/a(n) = Pi^6/945 = A013664. (End)
a(n) = Sum_{k=1..6} Eulerian(6, k)*binomial(n+6-k, 6), with Eulerian(6, k) = A008292(6, k) (the numbers are 1, 57, 302, 302, 57, 1) for n >= 0. Worpitzki's identity for powers of 6. See. e.g., Graham et al., eq. (6, 37) (using A173018, the row reversed version of A123125). - Wolfdieter Lang, Jul 17 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 31*zeta(6)/32 = 31*Pi^6/30240 (A275703). - Amiram Eldar, Oct 08 2020
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)^2/(6*Pi^2). (End)

Comments from 2010 - 2011 edited by M. F. Hasler, Jul 05 2024

A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

Original entry on oeis.org

1, 63, 45991, 2942695, 45982595359, 5109066151, 601081707598999, 38469080386820311, 252396118308232060471, 252395862211967012407, 447134922152359540530757327, 447134770212444455649757327, 2158234586764514215343657417779543, 308319185132349039219686748825649

Offset: 1

Alexander Adamchuk, Jan 16 2008

p divides a(p-1) for prime p > 2. a(n) is prime for n = {19, 47, 164, ...} = A136686.

Lim_{n -> infinity} a(n)/A334605(n) = A275703 = (31/32)*A013664. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = a(n)/A334605(n). - _Petros Hadjicostas_, May 07 2020

Harvey P. Dale, Table of n, a(n) for n = 1..382
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A013664, A058313, A119682, A120296, A136675, A136676.
Cf. A001008, A007406, A007408, A007410, A099828, A103345, A275703.
Cf. A136681, A136682, A136683, A136684, A136685, A136686, A334605 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^6, {k,1,n} ] ], {n,1,30} ]
Accumulate[Table[(-1)^(k+1)/k^6,{k,20}]]//Numerator (* Harvey P. Dale, Aug 21 2023 *)

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

A334605 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^6.

Original entry on oeis.org

1, 64, 46656, 2985984, 46656000000, 5184000000, 609892416000000, 39033114624000000, 256096265048064000000, 256096265048064000000, 453690155404813307904000000, 453690155404813307904000000, 2189875725319351517910798336000000

Offset: 1

Petros Hadjicostas, May 07 2020

Lim_{n -> infinity} A136677(n)/a(n) = A275703 = (31/32)*A013664.

			The first few fractions are: 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = A136677/A334605.

Cf. A013664, A136677 (numerators), A275703.

Denominator @ Accumulate[Table[(-1)^(k + 1)/k^6, {k, 1, 13}]] (* Amiram Eldar, May 07 2020 *)

a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^6)); \\ 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).

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A001014 Sixth powers: a(n) = n^6.

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A136677 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^6.

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A275710 Decimal expansion of the Dirichlet eta function at 7.

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A334605 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^6.

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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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A346927 Decimal expansion of the Dirichlet eta function at 10.

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