OFFSET
1,2
COMMENTS
It seems to me the sequence can never drop all the way to 1 again.
From Robert Israel, Oct 13 2015: (Start)
There are only finitely many n with a(n) = 1.
Such n correspond to solutions of the Diophantine equation x (2 x^2 + 3 x + 3) = z^3.
Since gcd(x, 2 x^2 + 3 x + 3) = 1 or 3, we get two cases:
if x is not divisible by 3, x = s^3, z = s^3 t^3 where 2 s^6 + 3 s^3 + 3 = t^3,
otherwise x = 9 s^3, z = 3 s t, where 54 s^6 + 9 s^3 + 1 = t^3.
The algebraic curves 2 s^6 + 3 s^3 + 3 = t^3 and 54 s^6 + 9 s^3 + 1 = t^3 both have genus 4, so by Faltings's theorem they have only finitely many rational solutions.
(End)
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..368
FORMULA
EXAMPLE
10 - 9 = 1 is the first number in the sequence because 10^3 + 9^3 = 12^3 + 1^3 and no other lower z produces a result.
MAPLE
Cubes:= {seq(x^3, x=2..10^4)}:
count:= 0:
for z from 1 to 10^4 do
s:= z^3+1;
M:= map(t -> s-t, select(`<`, Cubes, floor(s/2))) intersect Cubes;
for m in M do
count:= count+1;
y:= simplify(m^(1/3));
x:= simplify((s-m)^(1/3));
A[count]:= y-x;
od
od:
seq(A[i], i=1..count); # Robert Israel, Oct 13 2015
MATHEMATICA
y = 3; lst = {}; While[y < 100001, x = 2; While[x < y, z = (y^3 + x^3 - 1)^(1/3); If[IntegerQ[z], AppendTo[lst, {z, y, x, y - x}]; Print[{z, y, x, y - x}]]; x++]; y++]; Last@ Transpose@ Sort@ lst (* Robert G. Wilson v, Jul 21 2015 and modified Oct 14 2015 *)
PROG
(Python 3.x)
start = 12
end = 100000
print(start, end)
cubes = []
for i in range(0, end):
cubes.append(i**3)
for z in range(start, end):
target = cubes[z]+1
for x in range(2, z):
t = target-cubes[x]
y = int(t**(1/3)+.5)
if cubes[y] == t:
print(target, y, x, z, 1, y-x, '\a')
break
CROSSREFS
KEYWORD
nonn
AUTHOR
David Rabahy, Jul 21 2015
STATUS
approved