A118719 Cubes for which the digital root is also a cube.

0, 1, 8, 64, 125, 343, 512, 1000, 1331, 2197, 2744, 4096, 4913, 6859, 8000, 10648, 12167, 15625, 17576, 21952, 24389, 29791, 32768, 39304, 42875, 50653, 54872, 64000, 68921, 79507, 85184, 97336, 103823, 117649, 125000, 140608, 148877

Offset: 1

Author

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

Keywords

• base
• easy
• nonn

Comments

All cubes have a digital root 1,8 or 9. (except for the number 0) So this sequence contains all cubes with a digital root which is not 9.

This sequence is 0 union A016779 union A016791.

Examples

			64 is in the sequence because (1) it is a cube and (2) the digital root 1 is also a cube.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A000578, A002117, A010888, A116978.

Programs

Magma

[0] cat [(6*n+(-1)^n-9)^3 div 64: n in [2..37]];  // Bruno Berselli, May 05 2011

Mathematica

Join[{0}, Table[(3*k + {1, 2})^3, {k, 0, 15}] // Flatten] (* Amiram Eldar, Dec 19 2020 *)

PARI

a010888(n)=if(n, (n-1)%9+1)
lista(nn) = {for (n=0, nn, if (ispower(a010888(n^3), 3), print1(n^3, ", ")););} \\ Michel Marcus, Feb 18 2015

Formula

a(n) = (floor(3*n/2)-2)^3 for n >= 2. - Nathaniel Johnston, May 05 2011

G.f.: x^2*(1+7*x+53*x^2+40*x^3+53*x^4+7*x^5+x^6)/((1+x)^3*(1-x)^4). a(n) = A001651(n-1)^3 for n>1. - Bruno Berselli, May 05 2011

Sum_{n>=2} 1/a(n) = 26*zeta(3)/27. - Amiram Eldar, Dec 19 2020