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A023042
Numbers whose cube is the sum of three distinct nonnegative cubes.
14
6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 57, 58, 60, 63, 66, 67, 69, 70, 71, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 103, 105, 106, 108, 110, 111, 112, 113
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
STATUS
approved