

A060467


Value of z of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest z and smallest y, 0 <= x <= y <= z.


5



0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 11, 2, 1626, 2, 3, 3, 3, 16, 15584139827, 3, 3, 3, 3, 3, 2220422932, 8866128975287528, 3, 3, 3, 4, 4, 159380, 80538738812075974, 3, 8, 4, 3, 8, 31, 796, 61922712865, 3, 12, 3, 22, 4, 5, 5, 3, 4, 4, 4, 4, 5, 21, 4, 10, 4
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OFFSET

0,5


COMMENTS

Indexed by A060464.
Only primitive solutions where gcd(x,y,z) does not divide n are considered.
From the solution A060464(24) = 30 = 283059965^3  2218888517^3 + 2220422932^3 (smallest possible magnitudes according to A. Bogomolny), one has a(24) = 2220422932. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the last column of the table on Hisanori Mishima's web page.  M. F. Hasler, Nov 10 2015


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, Section D5.


LINKS

Table of n, a(n) for n=0..57.
A. Bogomolny, Finicky Diophantine Equations on cuttheknot.org, accessed Nov. 10, 2015.
A.S. Elsenhans, J. Jahnel, New sums of three cubes, Math. Comp. 78 (2009) 12271230.
K. Koyama, Y. Tsuruoka, H. Sekigawa, On searching for solutions of the Diophantine equation x^3+y^3+z^3=n, Math. Comp. 66 (1997) 841.
Hisanori Mishima, About n=x^3+y^3+z^3
Eric S. Rowland, Known families of integer solutions of x^3+y^3+z^3=n
A. Tyszka, A hypothetical upper bound for the solutions of a Diophantine equation with a finite number of solutions, arXiv:0901.2093 [math.NT], 20092014.


EXAMPLE

For n = 16 the smallest solution is 16 = (511)^3 + (1609)^3 + 1626^3, which gives the term 1626.
42 = 12602123297335631^3 + 80435758145817515^3 + (80538738812075974)^3 was found by Andrew Booker and Andrew Sutherland.
74 = 66229832190556^3 + 283450105697727^3 + (284650292555885)^3 was found by Sander Huisman.


MATHEMATICA

nmax = 29; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = {0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 7, 1, 511, 1, 1, 0, 1, 11, 2901096694, 1, 0, 0, 0, 1}; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; A060467 = Table[z /. ToRules[ Simplify[ r[A060464[[k]], A060465[[k]]] /. C[1] > 0]], {k, 1, Length[A060464]}] (* JeanFrançois Alcover, Jul 11 2012 *)


CROSSREFS

Cf. A060464, A060465, A060466.
Sequence in context: A064770 A322073 A332201 * A125918 A239202 A083533
Adjacent sequences: A060464 A060465 A060466 * A060468 A060469 A060470


KEYWORD

sign,nice,hard


AUTHOR

N. J. A. Sloane, Apr 10 2001


EXTENSIONS

In order to be consistent with A060465, where only primitive solutions are selected, a(18)=2 was replaced with 15584139827, by JeanFrançois Alcover, Jul 11 2012
Edited and a(24) added by M. F. Hasler, Nov 10 2015
a(25) from Tim Browning and further terms added by Charlie Neder, Mar 09 2019
More terms from Jinyuan Wang, Feb 14 2020
a(32) corrected by XU Pingya, May 11 2020


STATUS

approved



