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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 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[]]]}]]; 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 April 1 02:24 EDT 2020. Contains 333153 sequences. (Running on oeis4.)