A001017 Ninth powers: a(n) = n^9.

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

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

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)
• T. Lengyel, On divisibility of some power sums, INTEGERS, 7(2007), A41, 1-6.
• K. MacMillan and J. 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).

Crossrefs

Cf. A000578 (cubes), A013667 (zeta(9)), A256581.

Cf. A003391 - A004801 (sums of 2, ..., 12 positive 9th powers).

Programs

Magma

[n^9 : n in [0..40]]; // Wesley Ivan Hurt, Apr 01 2016

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 *)

PARI

vector(100, n, (n-1)^9) \\ Derek Orr, Aug 03 2014

PARI

A001017(n)=n^9 \\ M. F. Hasler, Jul 03 2025

Python

A001017 = lambda n: n**9 # M. F. Hasler, Jul 03 2025

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)

Extensions

More terms from James Sellers, Sep 19 2000