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Digits of an idempotent 6-adic number.
7

%I #43 Nov 01 2022 16:37:18

%S 4,4,3,5,0,2,4,3,3,3,0,4,0,0,4,1,4,2,4,3,0,0,0,5,0,3,0,0,0,2,4,1,2,2,

%T 5,1,3,3,1,5,4,2,2,4,1,5,3,5,4,3,0,3,1,5,3,2,2,5,2,1,0,0,3,0,0,1,2,3,

%U 2,4,0,1,0,1,5,4,4,5,1,3,5,4,2,5,4,0,5,1,2,0,5,4,5,3,1,5,2,1,3,3,2,3,3,5,3

%N Digits of an idempotent 6-adic number.

%C ( 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 A054869.

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

%H Kenny Lau, <a href="/A055620/b055620.txt">Table of n, a(n) for n = 0..9999</a>

%H V. deGuerre and R. A. Fairbairn, <a href="/A003226/a003226.pdf">Automorphic numbers</a>, J. Rec. Math., 1 (No. 3, 1968), 173-179.

%F If A is the 6-adic number, A == 4^(3^n) mod 6^n. - _Robert Dawson_, Oct 28 2022

%e (a(0) + a(1)*6 + a(2)*6^2 + a(3)*6^3)^2 == (a(0) + a(1)*6 + a(2)*6^2 + a(3)*6^3) mod 6^4 because 1478656 == 1216 (mod 1296).

%o (Python)

%o n=10000;res=pow((3**n+1)//2,n,3**n)*2**n

%o for i in range(n):print(i,res%6);res//=6

%o # _Kenny Lau_, Jun 09 2018

%o (PARI) first(p)=Vecrev(digits(lift(Mod(4,6^p)^3^p), 6)) \\ _Charles R Greathouse IV_, Nov 01 2022

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

%K nonn,base

%O 0,1

%A Paolo Dominici (pl.dm(AT)libero.it), Jun 04 2000