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A119598 Numbers that are repunits in four or more bases. 6
1, 31, 8191 (list; graph; refs; listen; history; text; internal format)
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 4-tuple (a,b,m,n) be a solution to the exponential Diophantine equation (a^m-1)/(a-1)=(b^n-1)/(b-1) with a>1, b>a, m>2 and n>2. Then (a^m-1)/(a-1) 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)/(x-1) = (y^n - 1)/(y-1), Pacific Journal of Mathematics 207:1 (2002), pp. 61-75.

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^3-1)/(n-1)], {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)/(n-1), {i, Length[pos]}]] (* T. D. Noe, Jun 07 2006 *)

CROSSREFS

Cf. A002275, A053696, A055129, A088323.

Cf. A053696 (numbers of the form (b^k-1)/(b-1)).

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

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Last modified August 20 06:04 EDT 2019. Contains 326139 sequences. (Running on oeis4.)