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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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A020938 A136360 A023483 * A128245 A117714 A245685
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)