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Numbers that can be written with a single digit in base 10 as well as in some base b<10.
1

%I #11 May 18 2021 06:29:44

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

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

%C 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

%e 777[base 10]=3333[base 6]

%o (Python)

%o from math import *

%o i=1

%o while i<(10**10-1)/9:

%o i=10*i+1

%o for m in range(1,10):

%o q=i*m

%o q2=q

%o for b in range(2,10):

%o restes=[]

%o q=q2

%o while q>0:

%o r=q%b

%o q=q//b

%o restes.append(r)

%o if restes==[restes[0]]*len(restes):

%o print(q2,restes,"en base ",b)

%K base,nonn,fini,full

%O 1,3

%A _Sébastien Dumortier_, Oct 10 2008