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

0,1

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.

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.

Formula

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

Example

(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).

Prog

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

Crossrefs

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

Sequence in context: A106147 A202393 A073321 * A072420 A258075 A286296

Adjacent sequences: A055617 A055618 A055619 * A055621 A055622 A055623

Keyword

nonn,base

Author

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

Status

approved