A001015 - OEIS

%I M5392 N2341 #80 Apr 13 2022 13:25:15

%S 0,1,128,2187,16384,78125,279936,823543,2097152,4782969,10000000,

%T 19487171,35831808,62748517,105413504,170859375,268435456,410338673,

%U 612220032,893871739,1280000000,1801088541,2494357888,3404825447,4586471424,6103515625,8031810176

%N Seventh powers: a(n) = n^7.

%C For n>0, (a(3*n-1)^7-a(2*n-1)^7-a(n)^7)/(7*(3*n-1)*(2*n-1)*n) = (2*A001106(n)+1)^2 (see Barisien reference, problem 173). - _Bruno Berselli_, Feb 01 2011

%C Number of the form a(n) + a(n+1) + ... + a(n+k) is never prime for all n, k>=0. This could be proved by the method indicated in the comment in A256581. - _Vladimir Shevelev_ and _Peter J. C. Moses_, Apr 04 2015

%D E.-N. Barisien, Supplemento al Periodico di Matematica, Raffaello Giusti Editore (Livorno), July 1913, p. 135 (Problem 173).

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001015/b001015.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F Multiplicative with a(p^e) = p^(7e). - _David W. Wilson_, Aug 01 2001

%F Totally multiplicative sequence with a(p) = p^7 for primes p. - _Jaroslav Krizek_, Nov 01 2009

%F 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) + 5040. - _Ant King_, Sep 24 2013

%F a(n) = n + Sum_{j=0..n-1}{k=1..6}binomial(7,k)*j^(7-k). - _Patrick J. McNab_, Mar 28 2016

%F G.f.: x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6)/(1-x)^8. See the Maple program. - _Wolfdieter Lang_, Oct 14 2016

%F From _Kolosov Petro_, Oct 22 2018: (Start)

%F a(n) = Sum_{k=1..n} A300785(n,k).

%F a(n) = Sum_{k=0..n-1} A300785(n,k). (End)

%F From _Amiram Eldar_, Oct 08 2020: (Start)

%F Sum_{n>=1} 1/a(n) = zeta(7) (A013665).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 63*zeta(7)/64 (A275710). (End)

%p A001015:=z*(1191*z^4+120*z^5+1191*z^2+2416*z^3+120*z+z^6+1)/(z-1)^8; # _Simon Plouffe_ in his 1992 dissertation; offset corrected by _M. F. Hasler_, Feb 01 2011

%t Table[n^7, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)

%o (Maxima) makelist(n^7,n,0,20); /* _Martin Ettl_, Jan 15 2013 */

%o (PARI) a(n)=n^7 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A000584, A013665, A275710, A300785.

%K nonn,easy,mult

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Sep 19 2000