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OFFSET
| 1,2
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COMMENTS
| 1 is a repunit in every base. 31 is a repunit in bases 1, 2, 5 and 30 8191 is a repunit in bases 1, 2, 90 and 8190 No others below 1,000,000. Can anyone extend the sequence?
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 (rayjchandler(AT)sbcglobal.net), 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 (noe(AT)sspectra.com), Jun 07 2006
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LINKS
| Eric Weisstein's World of Mathematics, Repunit.
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EXAMPLE
| E.g. a(2)=31 because 31 is a repunit in four bases:
31 (b1) = 1111111111111111111111111111111
31 (b2) = 11111
31 (b5) = 111
31 (b30) = 11.
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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 (noe(AT)sspectra.com), Jun 07 2006
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CROSSREFS
| Cf. A002275, A055129, A053696, A088323.
Cf. A053696 (numbers of the form (b^k-1)/(b-1)).
Cf. A145461 : bases 5 and 90 are 2 exceptions (Goormaghtigh's conjecture) [From Sebastien Dumortier (sdumortier(AT)ac-limoges.fr), Jan 24 2009]
Sequence in context: A059384 A136676 A135811 * A139295 A069451 A073099
Adjacent sequences: A119595 A119596 A119597 * A119599 A119600 A119601
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KEYWORD
| base,hard,more,nonn,bref
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AUTHOR
| Sergio Pimentel (ferdiego(AT)cox.net), Jun 01 2006
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EXTENSIONS
| Edited by Ray Chandler, Jun 08 2006
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