OFFSET
1,1
COMMENTS
Numbers w such that w^3 = x^3+y^3+z^3, x>y>z>=0, is soluble.
A226903(n) + 1 is an infinite subsequence parametrized by Shiraishi in 1826. - Jonathan Sondow, Jun 22 2013
Because of Fermat's Last Theorem, sequence lists numbers w such that w^3 = x^3+y^3+z^3, x>y>z>0, is soluble. In other words, z cannot be 0 because x and y are positive integers by definition of this sequence. - Altug Alkan, May 08 2016
This sequence is the same as numbers w such that w^3 = x^3+y^3+z^3, x>=y>=z>0, is soluble as Legendre showed that a^3+b^3=2*c^3 only has the trivial solutions a = b or a = -b (see Dickson's History of the Theory of Numbers, vol. II, p. 573). - Chai Wah Wu, May 13 2017
REFERENCES
Ya. I. Perelman, Algebra can be fun, pp. 142-143.
LINKS
Nathaniel Johnston and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (1..1500 from Nathaniel Johnston)
A. Russell and C. E. Gwyther, The Partition of Cubes, The Mathematical Gazette, 21 (1937), pp. 33-35.
EXAMPLE
20 belongs to the sequence as 20^3 = 7^3 + 14^3 + 17^3.
MAPLE
for w from 1 to 113 do for z from 0 to w-1 do bk:=0: for y from z+1 to w-1 do for x from y+((w+z) mod 2) to w-1 by 2 do if(x^3+y^3+z^3=w^3)then printf("%d, ", w); bk:=1: break: fi: od: if(bk=1)then break: fi: od: if(bk=1)then break: fi: od: od: # Nathaniel Johnston, Jun 22 2013
MATHEMATICA
lst={}; Do[Do[Do[Do[y=a^3+b^3+c^3; x=z^3; If[y<x, Break[], If[y==x, AppendTo[lst, z]]], {c, b-1, 0, -1}], {b, a-1, 0, -1}], {a, z-1, 0, -1}], {z, 2, 3*5!}]; Union@lst (* Vladimir Joseph Stephan Orlovsky, Apr 11 2010 *)
PROG
(PARI) has(n)=my(L=sqrtnint(n-1, 3)+1, U=sqrtnint(4*n, 3)); fordiv(n, m, if(L<=m && m<=U, my(ell=(m^2-n/m)/3); if(denominator(ell)==1 && issquare(m^2-4*ell), return(1)))); 0
list(lim)=my(v=List(), a3, t); lim\=1; for(a=2, sqrtint(lim\3), a3=a^3; for(b=if(a3>lim, sqrtnint(a3-lim-1, 3)+1, 1), a-1, t=a3-b^3; if(has(t), listput(v, a)))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved