A145461 Numbers that can be written with a single digit in base 10 as well as in some base b<10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 777

Offset: 1

Author

Sébastien Dumortier, Oct 10 2008

Keywords

• base
• nonn
• fini
• full

Comments

If a number is written in base 10 with a digit x and in base b with a digit y, then (b-1)*x*10^n - 9*y*b^m + (9*y - (b-1)*x) = 0. Varying parameters b=2,3,...,9; x=1,2,...,9; and y=1,2,...,b-1 give a finite number of equations. It is easy to find all solutions (w.r.t. n and m) of each equation or establish that there are none. In particular, for b=7, x=9, y=5, the equation is 54*10^n - 45*7^m - 9 = 0 or 6*10^n - 5*7^m - 1 = 0 that does not have solutions since the left hand side is not 0 modulo 5. It proves completeness and finiteness. - Max Alekseyev, Nov 06 2008

Examples

			777[base 10]=3333[base 6]
		

Programs

Python

from math import *
i=1
while i<(10**10-1)/9:
    i=10*i+1
    for m in range(1,10):
        q=i*m
        q2=q
        for b in range(2,10):
            restes=[]
            q=q2
            while q>0:
                r=q%b
                q=q//b
                restes.append(r)
            if restes==[restes[0]]*len(restes):
                print(q2,restes,"en base ",b)