cp's OEIS Frontend

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

%I A001014 M5330 N2318 #163 Jul 07 2025 00:46:14
%S A001014 0,1,64,729,4096,15625,46656,117649,262144,531441,1000000,1771561,
%T A001014 2985984,4826809,7529536,11390625,16777216,24137569,34012224,47045881,
%U A001014 64000000,85766121,113379904,148035889,191102976,244140625,308915776,387420489,481890304
%N A001014 Sixth powers: a(n) = n^6.
%C A001014 Numbers both square and cubic. - _Patrick De Geest_
%C A001014 Totally multiplicative sequence with a(p) = p^6 for prime p. - _Jaroslav Krizek_, Nov 01 2009
%C A001014 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
%C A001014 Note that Sum_{n>=1} 1/a(n) = Pi^6 / 945. - _Mohammad K. Azarian_, Nov 01 2011
%C A001014 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
%C A001014 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
%D A001014 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).
%D A001014 Granino A. Korn and Theresa M.Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
%D A001014 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001014 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001014 Franklin T. Adams-Watters, <a href="/A001014/b001014.txt">Table of n, a(n) for n = 0..500</a>
%H A001014 Henry Bottomley, <a href="/A001014/a001014.gif">Illustration of initial terms</a>
%H A001014 J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H A001014 Richard J. Mathar, <a href="https://arxiv.org/abs/1703.01677">Construction of Bhaskara pairs</a>, arXiv:1703.01677 [math.NT], 2017.
%H A001014 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001014 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001014 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F A001014 a(n) = A123866(n) + 1 = A002604(n) - 1.
%F A001014 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
%F A001014 Multiplicative with a(p^e) = p^(6e). - _David W. Wilson_, Aug 01 2001
%F A001014 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
%F A001014 From _Ant King_, Sep 23 2013: (Start)
%F A001014 Signature {7, -21, 35, -35, 21, -7, 1}.
%F A001014 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)
%F A001014 a(n) == 1 (mod 7) if gcd(n, 7) = 1, otherwise a(n) == 0 (mod 7). See A109720. - _Jake Lawrence_, May 28 2016
%F A001014 From _Ilya Gutkovskiy_, Jul 06 2016: (Start)
%F A001014 Dirichlet g.f.: zeta(s-6).
%F A001014 Sum_{n>=1} 1/a(n) = Pi^6/945 = A013664. (End)
%F A001014 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
%F A001014 Sum_{n>=1} (-1)^(n+1)/a(n) = 31*zeta(6)/32 = 31*Pi^6/30240 (A275703). - _Amiram Eldar_, Oct 08 2020
%F A001014 From _Amiram Eldar_, Jan 20 2021: (Start)
%F A001014 Product_{n>=1} (1 + 1/a(n)) = (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).
%F A001014 Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)^2/(6*Pi^2). (End)
%e A001014 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.
%t A001014 Table[n^6, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%o A001014 (Haskell)
%o A001014 a001014 n = a001014_list !! n
%o A001014 a001014_list = map (^ 6) [0..] -- _Reinhard Zumkeller_, Dec 04 2011
%o A001014 (Maxima) A001014(n):=n^6$
%o A001014 makelist(A001014(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A001014 (PARI) A001014(n)=n^6 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A001014 (Python) A001014 = lambda n: n**6 # _M. F. Hasler_, Jul 03 2025
%Y A001014 Subsequence of A201217.
%Y A001014 Cf. A000540 (partial sums), A022522 (first differences), A008292.
%Y A001014 Intersection of A000290 (squares) and A000578 (cubes).
%Y A001014 Cf. A002604 (n^6+1), A123866 (n^6-1), A013664 (zeta(6)), A275703 (eta(6)).
%Y A001014 Cf. A003358 - A003368 (sums of 2, ..., 12 positive sixth powers).
%K A001014 nonn,easy,mult
%O A001014 0,3
%A A001014 _N. J. A. Sloane_
%E A001014 Comments from 2010 - 2011 edited by _M. F. Hasler_, Jul 05 2024