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A023042 Numbers whose cube is the sum of three distinct nonnegative cubes. 11
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

Cf. A001235, A003072, A114923, A225908, A226903.

Sequence in context: A020938 A136360 A023483 * A128245 A117714 A245685

Adjacent sequences:  A023039 A023040 A023041 * A023043 A023044 A023045

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)