1,3

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. (Alekseyev)

777[base 10]=3333[base 6]

(Python) from math import *

.i=1 while i<(10**1000-1)/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

Sequence in context: A069747 A124107 A112014 this_sequence A075154 A070938 A070594

Adjacent sequences: A145458 A145459 A145460 this_sequence A145462 A145463 A145464

base,nonn,full,fini

Sebastien Dumortier (sdumortier(AT)ac-limoges.fr), Oct 10 2008

Completeness and finiteness established by Max Alekseyev (maxale(AT)gmail.com), Nov 06 2008