A008456 12th powers: a(n) = n^12.

0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, 23298085122481, 56693912375296, 129746337890625, 281474976710656, 582622237229761

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

Numbers which are square, cubic and quartic. - Doug Bell, Jun 03 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

a(n) = A123868(n) + 1.

Cf. A000290 (squares), A000578 (cubes), A000583 (4th powers), A001014 (6th powers), A008454 (10th powers), A008455 (11th powers), A010801 (13th powers).

Cf. A013670 (zeta(12)).

Programs

Magma

[n^12: n in [0..30]]; // Vincenzo Librandi, May 06 2011

Mathematica

Range[0,30]^12 (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

PARI

A008456(n)=n^12 \\ M. F. Hasler, Jul 03 2025

Python

A008456 = lambda n: n**12 # M. F. Hasler, Jul 03 2025

Formula

Multiplicative with a(p^e) = p^(12*e). - David W. Wilson, Aug 01 2001

a(n) = A000290(n)^6 = A000578(n)^4 = A000583(n)^3 = A001014(n)^2. - Doug Bell, Jun 03 2017

From Amiram Eldar, Oct 08 2020: (Start)

Sum_{n>=1} 1/a(n) = zeta(12) = 691*Pi^12/638512875 (A013670).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2047*zeta(12)/2048 = 1414477*Pi^12/1307674368000. (End)

a(n) = 13*a(n-1)-78*a(n-2)+286*a(n-3)-715*a(n-4)+1287*a(n-5)-1716*a(n-6)+1716*a(n-7)-1287*a(n-8)+715*a(n-9)-286*a(n-10)+78*a(n-11)-13*a(n-12)+a(n-13). - Wesley Ivan Hurt, Dec 02 2021

Intersection of A000578 and A000583; i.e., cubes and 4th powers. - M. F. Hasler, Jul 03 2025