A054869 Digits of an idempotent 6-adic number.

3, 1, 2, 0, 5, 3, 1, 2, 2, 2, 5, 1, 5, 5, 1, 4, 1, 3, 1, 2, 5, 5, 5, 0, 5, 2, 5, 5, 5, 3, 1, 4, 3, 3, 0, 4, 2, 2, 4, 0, 1, 3, 3, 1, 4, 0, 2, 0, 1, 2, 5, 2, 4, 0, 2, 3, 3, 0, 3, 4, 5, 5, 2, 5, 5, 4, 3, 2, 3, 1, 5, 4, 5, 4, 0, 1, 1, 0, 4, 2, 0, 1, 3, 0, 1, 5, 0, 4, 3, 5, 0, 1, 0, 2, 4, 0, 3, 4, 2

Offset: 0

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

• nonn
• base

Comments

( a(0) + a(1)*6 + a(2)*6^2 + ... )^k = a(0) + a(1)*6 + a(2)*6^2 + ... for each k. Apart from 0 and 1, in base 6 there are only 2 numbers with this property. For the other see A055620.

References

• V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Links

• Kenny Lau, Table of n, a(n) for n = 0..9999
• V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179

Crossrefs

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

Programs

Python

n=10000;res=1-pow((3**n+1)//2,n,3**n)*2**n
for i in range(n):print(i,res%6);res//=6
# Kenny Lau, Jun 09 2018

Formula

a(n) == 3^(2^n) (mod 6^n). - Robert Dawson, Oct 28 2022