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
Examples
121 = (11111)_3, 133 = (111)_11 = (77)_18.
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.
Crossrefs
Programs
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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 -
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
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