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A145461 Numbers that can be written with a single digit in base 10 as well as in some base b<10. 1
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 777 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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. (Alekseyev)

LINKS

Table of n, a(n) for n=1..11.

EXAMPLE

777[base 10]=3333[base 6]

PROG

(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

CROSSREFS

Sequence in context: A124107 A257275 A112014 * A075154 A187924 A070938

Adjacent sequences:  A145458 A145459 A145460 * A145462 A145463 A145464

KEYWORD

base,nonn,full,fini

AUTHOR

S├ębastien Dumortier, Oct 10 2008

EXTENSIONS

Completeness and finiteness established by Max Alekseyev, Nov 06 2008

STATUS

approved

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Last modified July 9 18:08 EDT 2020. Contains 335545 sequences. (Running on oeis4.)