OFFSET
1,1
COMMENTS
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).
Only three known perfect powers belong to this sequence: 121, 343 and 400 (A208242).
Except for 121, each term of this sequence have also at least one Brazilian representation with only 2 digits.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Yann Bugeaud and T. N. Shorey, On the Diophantine Equation (x^m - 1)/(x-1) = (y^n - 1)/(y-1), Pacific Journal of Mathematics, Vol. 207, No 1, November 2002.
Sean A. Irvine, Java program (github)
Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, 6th World Conference on 21st Century Mathematics 2013, Lahore, p. 14 (Goormaghtigh conjecture).
Wikipedia, Goormatigh conjecture.
EXAMPLE
121 = (11111)_3, 133 = (111)_11 = (77)_18.
MAPLE
N:= 3000:
Res:= NULL:
for m from 2 while 1+m+m^2 <= N do
for k from 2 do
v:= (m^(k+1)-1)/(m-1);
if v > N then break fi;
if not isprime(v) then Res:= Res, v fi
od od:
sort(convert({Res}, list)); # Robert Israel, May 13 2019
PROG
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Bernard Schott, May 12 2019
STATUS
approved