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

15, 21, 40, 57, 63, 85, 91, 111, 121, 133, 156, 183, 255, 259, 273, 341, 343, 364, 381, 400, 507, 511, 553, 585, 651, 703, 781, 813, 820, 871, 931, 993, 1023, 1057, 1111, 1191, 1261, 1333, 1365, 1407, 1464, 1555, 1561, 1641, 1807, 1885, 1893, 1981, 2047, 2071, 2163, 2257, 2353

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

Complement of A085104 with respect to A053696.

Intersection of A053696 and A220571.

Cf. A119598, A208242, A325659.

Sequence in context: A349096 A063175 A367103 * A331628 A171569 A247021

Adjacent sequences: A325655 A325656 A325657 * A325659 A325660 A325661

Keyword

nonn,base

Author

Bernard Schott, May 12 2019

Status

approved