

A119598


Numbers that are repunits in four or more bases.


10




OFFSET

1,2


COMMENTS

Except for first term, numbers which can be represented as a string of three or more 1's in a base >=2 in more than one way; subset of A053696.
No more terms less than 2^44 = 17592186044416.  Ray Chandler, Jun 08 2006
Let the 4tuple (a,b,m,n) be a solution to the exponential Diophantine equation (a^m1)/(a1)=(b^n1)/(b1) with a>1, b>a, m>2 and n>2. Then (a^m1)/(a1) is in this sequence. The terms 31 and 8191 correspond to the solutions (2,5,5,3) and (2,90,13,3), respectively. No other solutions with n=3 and b<10^5. The Mathematica code finds repunits in increasing order and prints solutions.  T. D. Noe, Jun 07 2006
Following the Goormaghtigh conjecture (Links), 31 and 8191 which are both Mersenne numbers, are the only primes which are Brazilian in two different bases.  Bernard Schott, Jun 25 2013


LINKS

Table of n, a(n) for n=1..3.
Y. Bugeaud and T. N. Shorey, On the diophantine equation (x^m  1)/(x1) = (y^n  1)/(y1), Pacific Journal of Mathematics 207:1 (2002), pp. 6175.
Eric Weisstein's World of Mathematics, Repunit
Wikipedia, Goormaghtigh conjecture


EXAMPLE

a(1)=1 is a repunit in every base. a(2)=31 is a repunit in bases 1, 2, 5 and 30. a(3)=8191 is a repunit in bases 1, 2, 90 and 8190.
31 and 8191 are Brazilian numbers in two different bases:
31 = 11111_2 = 111_5,
8191 = 1111111111111_2 = 111_90.


MATHEMATICA

fQ[n_] := Block[{d = Rest@Divisors[n  1]}, Length@d > 2 && Length@Select[IntegerDigits[n, d], Union@# == {1} &] > 2]; Do[ If[ fQ@n, Print@n], {n, 10^8/3}] (* Robert G. Wilson v *)
nn=1000; pow=Table[3, {nn}]; t=Table[If[n==1, Infinity, (n^31)/(n1)], {n, nn}]; While[pos=Flatten[Position[t, Min[t]]]; !MemberQ[pos, nn], If[Length[pos]>1, Print[{pos, pow[[pos]], t[[pos[[1]]]]}]]; Do[n=pos[[i]]; pow[[n]]++; t[[n]]=(n^pow[[n]]1)/(n1), {i, Length[pos]}]] (* T. D. Noe, Jun 07 2006 *)


PROG

(Python)
def isrep(n, b):
while n >= b:
n, r = divmod(n, b)
if r != 1: return False
return n == 1
def agen():
yield 1
n = 2
while True:
reps = 2 # n is a repunit in bases 1 and n1
for b in range(2, n1):
if isrep(n, b): reps += 1
if reps == 4: yield n; break
n += 1
for m in agen(): print(m) # Michael S. Branicky, Jan 31 2021


CROSSREFS

Cf. A002275, A053696, A055129, A088323.
Cf. A053696 (numbers of the form (b^k1)/(b1)).
Cf. A145461: bases 5 and 90 are 2 exceptions (Goormaghtigh's conjecture).
Cf. A085104 (Brazilian primes).
Sequence in context: A059384 A136676 A135811 * A139295 A261947 A069451
Adjacent sequences: A119595 A119596 A119597 * A119599 A119600 A119601


KEYWORD

base,hard,more,nonn,bref


AUTHOR

Sergio Pimentel, Jun 01 2006


EXTENSIONS

Edited by Ray Chandler, Jun 08 2006


STATUS

approved



