A001016 Eighth powers: a(n) = n^8.

0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281, 110075314176

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

Besides the first term, this sequence lists the denominators in Pi^8/9450 = 1 + 1/256 + 1/6561 + 1/65536 + 1/390625 + 1/1679616 + ... - Mohammad K. Azarian, Nov 01 2011, edited by M. F. Hasler, Jul 03 2025

For n > 0, a(n) is the largest number k such that k + n^4 divides k^2 + n^4. - Derek Orr, Oct 01 2014

Fourth powers of squares and squares of 4th powers. Squares composed with themselves twice. - Wesley Ivan Hurt, Apr 01 2016

References

• Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
• 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

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Crossrefs

Cf. A000290 (squares), A000583 (fourth powers), A001014 - A001017 (6th - 9th powers), A008454 (10th powers), A010801 (13th powers).

Cf. A000542 (partial sums), A022524 (first differences), A013666 (zeta(8)).

Cf. A003380 - A003390 (sums of 2, ..., 12 eighth powers).

Programs

Magma

[n^8 : n in [0..50]]; // Wesley Ivan Hurt, Apr 01 2016

Maple

A001016:=n->n^8: seq(A001016(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2016

Mathematica

Table[n^8, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)

Maxima

A001016(n):=n^8$
makelist(A001016(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

A001016(n)=n^8 \\ Charles R Greathouse IV, Sep 24 2015

Python

A001016 = lambda n: n**8 # M. F. Hasler, Jul 03 2025

Formula

Multiplicative with a(p^e) = p^(8e). - David W. Wilson, Aug 01 2001

Totally multiplicative sequence with a(p) = p^8 for primes p. - Jaroslav Krizek, Nov 01 2009

G.f.: -x*(1+x)*(x^6+246*x^5+4047*x^4+11572*x^3+4047*x^2+246*x+1)/(x-1)^9. - R. J. Mathar, Jan 07 2011

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 40320. - Ant King, Sep 24 2013

From Wesley Ivan Hurt, Apr 01 2016: (Start)

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n > 8.

a(n) = A000290(n)^4 = A000290(A000290(A000290(n))).

a(n) = A000583(n)^2. (End)

From Amiram Eldar, Oct 08 2020: (Start)

Sum_{n>=1} 1/a(n) = zeta(8) = Pi^8/9450 (A013666).

Sum_{n>=1} (-1)^(n+1)/a(n) = 127*zeta(8)/128 = 127*Pi^8/1209600. (End)

E.g.f.: exp(x)*x*(1 + 127*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7). - Stefano Spezia, Jul 29 2022

Extensions

More terms from James Sellers, Sep 19 2000