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URL: https://oeis.org/A001017

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A001017
Ninth powers: a(n) = n^9.
52
0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, 38443359375, 68719476736, 118587876497, 198359290368, 322687697779, 512000000000, 794280046581, 1207269217792
OFFSET
0,3
COMMENTS
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
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
Composition of the cubes with themselves. - Wesley Ivan Hurt, Apr 01 2016
REFERENCES
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).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
Tamás Lengyel, On divisibility of some power sums, INTEGERS, 7(2007), A41, 1-6.
Kieren MacMillan and Jonathan Sondow, Divisibility of power sums and the generalized Erdős-Moser equation, arXiv:1010.2275 [math.NT], 2010-2011.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
Multiplicative with a(p^e) = p^(9e). - David W. Wilson, Aug 01 2001
Totally multiplicative sequence with a(p) = p^9 for primes p. - Jaroslav Krizek, Nov 01 2009
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
a(n) = A000578(n)^3. - Wesley Ivan Hurt, Apr 01 2016
From Amiram Eldar, Oct 08 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(9) (A013667).
Sum_{n>=1} (-1)^(n+1)/a(n) = 255*zeta(9)/256. (End)
From Enrique Navarrete, Feb 02 2026: (Start)
Dirichlet g.f.: zeta(s-9).
E.g.f.: x*(1 + 255*x + 3025*x^2 + 7770*x^3 + 6951*x^4 + 2646*x^5 + 462*x^6 + 36*x^7 + x^8)*exp(x). (End)
MAPLE
A001017:=n->n^9: seq(A001017(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2016
MATHEMATICA
Table[n^9, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
Range[0, 30]^9 (* Wesley Ivan Hurt, Apr 01 2016 *)
PROG
(PARI) vector(100, n, (n-1)^9) \\ Derek Orr, Aug 03 2014
(PARI) A001017(n)=n^9 \\ M. F. Hasler, Jul 03 2025
(Magma) [n^9 : n in [0..40]]; // Wesley Ivan Hurt, Apr 01 2016
(Python) A001017 = lambda n: n**9 # M. F. Hasler, Jul 03 2025
CROSSREFS
Cf. A000578 (cubes), A013667 (zeta(9)), A256581.
Cf. A003391 - A004801 (sums of 2, ..., 12 positive 9th powers).
Partial sums are A007487.
First differences are A022525.
Sequence in context: A321833 A351197 A017682 * A352055 A351607 A343289
KEYWORD
nonn,mult,easy
EXTENSIONS
More terms from James Sellers, Sep 19 2000
STATUS
approved