cp's OEIS Frontend

A325658 Brazilian composites of the form 1 + b + b^2 + b^3 + ... + b^k, b > 1, k > 1.

%I A325658 #36 May 20 2019 15:33:32
%S A325658 15,21,40,57,63,85,91,111,121,133,156,183,255,259,273,341,343,364,381,
%T A325658 400,507,511,553,585,651,703,781,813,820,871,931,993,1023,1057,1111,
%U A325658 1191,1261,1333,1365,1407,1464,1555,1561,1641,1807,1885,1893,1981,2047,2071,2163,2257,2353
%N A325658 Brazilian composites of the form 1 + b + b^2 + b^3 + ... + b^k, b > 1, k > 1.
%C A325658 Composites that are repunits in base b >= 2 with three or more digits. If the Goormaghtigh conjecture is true, there are no composite numbers which can be represented as a string of three or more 1's in a base >= 2 in more than one way (A119598).
%C A325658 Only three known perfect powers belong to this sequence: 121, 343 and 400 (A208242).
%C A325658 Except for 121, each term of this sequence have also at least one Brazilian representation with only 2 digits.
%H A325658 Robert Israel, <a href="/A325658/b325658.txt">Table of n, a(n) for n = 1..10000</a>
%H A325658 Yann Bugeaud and T. N. Shorey, <a href="https://msp.org/pjm/2002/207-1/pjm-v207-n1-p04-s.pdf">On the Diophantine Equation (x^m - 1)/(x-1) = (y^n - 1)/(y-1)</a>, Pacific Journal of Mathematics, Vol. 207, No 1, November 2002.
%H A325658 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a325/A325658.java">Java program</a> (github)
%H A325658 Michel Waldschmidt, <a href="https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf">Lecture on the abc conjecture and some of its consequences</a>, 6th World Conference on 21st Century Mathematics 2013, Lahore, p. 14 (Goormaghtigh conjecture).
%H A325658 Wikipedia, <a href="https://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormatigh conjecture</a>.
%e A325658 121 = (11111)_3, 133 = (111)_11 = (77)_18.
%p A325658 N:= 3000:
%p A325658 Res:= NULL:
%p A325658 for m from 2 while 1+m+m^2 <= N do
%p A325658   for k from 2 do
%p A325658     v:= (m^(k+1)-1)/(m-1);
%p A325658     if v > N then break fi;
%p A325658     if not isprime(v) then  Res:= Res, v fi
%p A325658 od od:
%p A325658 sort(convert({Res},list)); # _Robert Israel_, May 13 2019
%o A325658 (PARI) lista(nn) = {forcomposite(n=1, nn, for(b=2, sqrtint(n), my(d=digits(n, b), sd=Set(d)); if ((#d >= 3) && (#sd == 1) && (sd[1] == 1), print1(n, ", "); break);););} \\ _Michel Marcus_, May 18 2019
%Y A325658 Complement of A085104 with respect to A053696.
%Y A325658 Intersection of A053696 and A220571.
%Y A325658 Cf. A119598, A208242, A325659.
%K A325658 nonn,base
%O A325658 1,1
%A A325658 _Bernard Schott_, May 12 2019