A001017 - OEIS

%I M5459 N2368 #73 Oct 27 2023 19:09:05

%S 0,1,512,19683,262144,1953125,10077696,40353607,134217728,387420489,

%T 1000000000,2357947691,5159780352,10604499373,20661046784,38443359375,

%U 68719476736,118587876497,198359290368,322687697779,512000000000,794280046581,1207269217792

%N Ninth powers: a(n) = n^9.

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

%C A generalization. Using modified Lengyel's 2007 ideas one can prove that, for every odd r>=3, every number of the form n^r + (n+1)^r + ... + (n+k)^r is nonprime. - _Vladimir Shevelev_, Apr 04 2015

%C Composition of the cubes with themselves. - _Wesley Ivan Hurt_, Apr 01 2016

%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 Michael De Vlieger, <a href="/A001017/b001017.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)

%H T. Lengyel, <a href="http://www.emis.de/journals/INTEGERS/papers/h41/h41.Abstract.html">On divisibility of some power sums</a>, INTEGERS, 7(2007), A41, 1-6.

%H K. MacMillan and J. Sondow, <a href="http://arxiv.org/abs/1010.2275">Divisibility of power sums and the generalized Erdős-Moser equation</a>, arXiv:1010.2275 [math.NT], 2010-2011.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F Multiplicative with a(p^e) = p^(9e). - _David W. Wilson_, Aug 01 2001

%F Totally multiplicative sequence with a(p) = p^9 for primes p. - _Jaroslav Krizek_, Nov 01 2009

%F G.f.:  x*(1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(x-1)^10. - _R. J. Mathar_, Jan 07 2011

%F a(n) = A000578(n)^3. - _Wesley Ivan Hurt_, Apr 01 2016

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

%F Sum_{n>=1} 1/a(n) = zeta(9) (A013667).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 255*zeta(9)/256. (End)

%p A001017:=n->n^9: seq(A001017(n), n=0..30); # _Wesley Ivan Hurt_, Apr 01 2016

%t Table[n^9, {n, 0, 20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)

%t Range[0, 30]^9 (* _Wesley Ivan Hurt_, Apr 01 2016 *)

%o (PARI) vector(100, n, (n-1)^9) \\ _Derek Orr_, Aug 03 2014

%o (Magma) [n^9 : n in [0..40]]; // _Wesley Ivan Hurt_, Apr 01 2016

%Y Cf. A000578 (cubes), A013667, A256581.

%K nonn,mult,easy

%O 0,3

%A _N. J. A. Sloane_

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