OFFSET
1,1
COMMENTS
Numbers n such that n^3+1 is expressible as the sum of two nonzero cubes (both greater than 1).
Values of z associated with A050794.
Sequence is infinite. One subsequence is (from x = 1 + 9 m^3, y = 9 m^4, z = 3*m*(3*m^3 + 1), x^3 + y^3 = z^3 + 1): z(m) = 3*m*(3*m^3 + 1) = {12, 150, 738, 2316, 5640, 11682, 21630, 36888, 59076, 90030, ...} = a (1, 3, 6, 10, 13, 17, 20, 23, 30, 37, ...). - Zak Seidov, Sep 16 2013
Numbers n such that n^3+1 is a member of A001235. - Altug Alkan, May 09 2016
REFERENCES
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
LINKS
Lewis Mammel, Table of n, a(n) for n = 1..368
Noam Elkies, Rational points near curves and small nonzero |x^3-y^2| via lattice reduction, arXiv:math/0005139 [math.NT], 2000.
S. Ramanujan, Question 681, J. Ind. Math. Soc.
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers
EXAMPLE
12 is a term because 10^3 + 9^3 = 12^3 + 1 (= 1729).
2316 is in the sequence because 577^3 + 2304^3 = 2316^3 + 1.
MATHEMATICA
r[z_] := Reduce[ 1 < x < y < z && x^3 + y^3 == z^3 + 1, {x, y}, Integers]; z = 4; A050791 = {}; While[z < 10^4, If[r[z] =!= False, Print[z]; AppendTo[A050791, z]]; z++]; A050791 (* Jean-François Alcover, Dec 27 2011 *)
PROG
(PARI) is(n)=if(n<2, return(0)); my(c3=n^3); for(a=2, sqrtnint(c3-5, 3), if(ispower(c3-1-a^3, 3), return(1))); 0 \\ Charles R Greathouse IV, Oct 26 2014
(PARI) T=thueinit('x^3+1); is(n)=n>8&&#select(v->min(v[1], v[2])>1, thue(T, n^3+1))>0 \\ Charles R Greathouse IV, Oct 26 2014
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Patrick De Geest, Sep 15 1999
EXTENSIONS
More terms from Michel ten Voorde
Extended through 47584 by Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Edited by N. J. A. Sloane, May 08 2007
STATUS
approved