A332201 -id:A332201 - cp's OEIS frontend

A060465 Value of x 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|.

0, 0, 0, 1, -1, 0, 0, 0, 1, -2, 7, -1, -511, 1, -1, 0, 1, -11, -2901096694, -1, 0, 0, 0, 1, -283059965, -2736111468807040, -1, 0, 1, 0, 1, 117367, 12602123297335631, 2, -5, 2, -2, 6, -23, 602, 23961292454, -1, -7, 1, -11, 1, -1, 0, 2, 0, 0, 0, 1, 2, 11, -1, 7, 1

Offset: 0

N. J. A. Sloane, Apr 10 2001

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) = -283059965. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the first column of the table on Hisanori Mishima's web page. - M. F. Hasler, Nov 10 2015
In 2019 Brooker found a solution for n = 33 (see A332201 and references there) and later in the same year for n = 42, using the collaborative "Charity Engine". It would be nice to have information on how far it is established that these solutions are the smallest possible. - M. F. Hasler, Feb 24 2020

			For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -511.
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.

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, New York, 2004, Section D5, 231-234.

D. J. Bernstein, Three cubes
A. Bogomolny, Finicky Diophantine Equations on cut-the-knot.org, accessed Nov. 10, 2015.
B. Conn, L. Vaserstein, On sums of three integral cubes, Contemp. Math 166 (1994) MR1284068
V. L. Gardiner, R. B. Lazarus, P. R. Stein, Solutions of the diophantine equation x^3+y^3=z^3-d, Math. Comp. 18 (1964) 408-413.
D.R. Heath-Brown, W.M. Lioen and H.J.J. te Riele on Solving the Diophantine Equation x3 + y3 + z3 = k on a Vector Computer
J. C. P. Miller, M. F. C. Woollett, Solutions of the Diophantine Equation x^3+y^3+z^3=k, J. Lond. Math. Soc. 30 (1) (1955) 101-110.
Hisanori Mishima, About n=x^3+y^3+z^3

Cf. A060464, A060466, A060467, A173515.

(* this program is not convenient for hard cases *) nmax = 29; xmin[] = 0; xmax[] = 20; xmin[16] = 500; xmax[16] = 600; xmin[24] = 2901096600; xmax[24] = 2901096700; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; r[n_ /; IntegerQ[n^(1/3)]] := {0, 0, n^(1/3)}; mySort = Sort[#1, Which[Abs[#1[[3]]] <= Abs[#2[[3]]], True, Abs[#1[[3]]] == Abs[#2[[3]]], If[Abs[#1[[2]]] <= Abs[#2[[2]]], True, False], True, False] & ] & ; rep := {x_, y_, z_} /; (x + y == 0 && x > 0) :> {-x, -y, z}; r[n_] := Reap[Do[ sp = r[n, x] /. C[1] -> 1; If[sp =!= False, xyz = {x, y, z} /. {ToRules[sp]} /. rep; If[GCD @@ Flatten[{n, xyz}] == 1, Sow[xyz]]]; sn = r[n, -x] /. C[1] -> 1; If[sn =!= False, xyz = {-x, y, z} /. {ToRules[sn]} /. rep; If[GCD @@ Flatten[{n, xyz}] == 1, Sow[xyz]]], {x, xmin[n], xmax[n]}]][[2, 1]] // Flatten[#, 1] & // mySort // First; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = Table[xyz = r[n]; Print[ " n = ", n, " {x,y,z} = ", xyz]; xyz[[1]], {n, A060464}] (* Jean-François Alcover, Jul 10 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 13 2020

A246869 Cube root of the smallest of the largest absolute values of parts of the partitions of n into four cubes, or -1 if no such partition exists.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 11, 2, 2, 2, 2, 2, 3, 3, 3, 16, 2, 2, 2, 3, 3, 3, 3, 3, 52, 2, 3, 3, 3, 3, 3, 3, 4, 4, 8, 3, 3, 3, 3, 3, 3, 4, 4, 49, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 3, 3, 3, 4, 4, 4, 4, 5, 5, 3, 4, 4, 3, 4, 4, 11, 5, 8, 4, 3, 3, 3, 4, 4

Offset: 0

David S. Newman, Sep 05 2014

nonn

It is not known if every integer can be written as the sum of four cubes, but it is true at least up to 1000 by computer search.
For each partition of n into four cubes (positive, negative, or zero) choose the largest part in absolute value. a(n) is the cube root of the smallest such largest part over all such partitions.
If there is no partition of n into four cubes, then a(n) = -1.
There is an interesting correlation with A332201 (sum of three cubes problem) whose nonzero absolute values coincide with a(n+1) up to n=30. - M. F. Hasler, Feb 10 2020

			The partition of 13 into 1^3+7^3+10^3+(-11)^3 has a part 11^3 in absolute value. Any other partition of 13 into four cubes has a part larger than 11^3 in absolute value. Thus a(13) = 11.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A243113.

b:= proc(n, i, t) n=0 or t*i^3>=n and (b(n, i-1, t)
      or b(n+i^3, i, t-1) or b(abs(n-i^3), i, t-1))
    end:
a:= proc(n) local k; for k from 0
      do if b(n, k, 4) then return k fi od
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 05 2014

b[n_, i_, t_] := b[n, i, t] = n == 0 || t i^3 >= n && (b[n, i - 1, t] || b[n + i^3, i, t - 1] || b[Abs[n - i^3], i, t - 1]);
a[n_] := Module[{k}, For[k = 0, True, k++, If[b[n, k, 4], Return[k]]]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

More terms from Alois P. Heinz, Sep 05 2014

cp's OEIS Frontend

A060465 Value of x 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|.

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