OFFSET
1,1
COMMENTS
n^3 - 1 is expressible as the sum of two distinct positive cubes. [corrected by Altug Alkan, Apr 11 2016]
The subsequence of primes in the sequence begins: 577, 38239, 69709. - Jonathan Vos Post, May 13 2010
Sequence is infinite. One subsequence is b (m) = 9 m^4 = {9, 144, 729, 2304, 5625, 11664, 21609, 36864, 59049, ...} = a (1, 2, 6, 10, 13, 17, 21, 24, 31, ...). - Zak Seidov, Sep 16 2013
REFERENCES
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, On number "729", p. 147.
LINKS
Jean-François Alcover and Charles R Greathouse IV, Table of n, a(n) for n = 1..104 (first 60 terms from Alcover)
Eric Weisstein's World of Mathematics, Diophantine Equation - 3rd Powers
EXAMPLE
2304 is in the sequence because 575^3 + 2292^3 = 2304^3 - 1.
MAPLE
N:= 10000: # to get all entries <= N
P:= proc(r)
local dcands, xs;
dcands:= select(d -> issqr(-3*d^4+12*d*r), numtheory[divisors](r));
xs:= map(d -> [solve(d^2-3*d*x+3*x^2-r/d, x)], dcands);
select(p -> p[1]<>p[2], select(type, xs, list(posint)));
end proc:
select(z -> nops(P(z^3-1))>0, [$1..N]); # Robert Israel, Jun 09 2014
MATHEMATICA
r[z_] := Reduce[1 < x < y < z && x^3 + y^3 == z^3 - 1, {x, y}, Integers]; Reap[z = 4; While[z < 10^5, rz = r[z]; If[rz =!= False, Print[xyz = {x, y, z} /. ToRules[rz]]; Sow[xyz[[3]]]]; z++]][[2, 1]] (* Jean-François Alcover, Dec 27 2011, updated Feb 11 2014 *)
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 Jud McCranie, Dec 25 2000
More terms from Don Reble, Nov 29 2001
Definition corrected by Robert Israel, Jun 09 2014
STATUS
approved