A055620 Digits of an idempotent 6-adic number.

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, 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, 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

Offset: 0

Author

Paolo Dominici (pl.dm(AT)libero.it), Jun 04 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 A054869.

Examples

			(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).
		

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, J. 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

PARI

first(p)=Vecrev(digits(lift(Mod(4,6^p)^3^p), 6)) \\ Charles R Greathouse IV, Nov 01 2022

Python

n=10000;res=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

If A is the 6-adic number, A == 4^(3^n) mod 6^n. - Robert Dawson, Oct 28 2022