A060467 -id:A060467 - cp's OEIS frontend

A060464 Numbers that are not congruent to 4 or 5 mod 9.

0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 87, 88, 89, 90, 91

Offset: 1

N. J. A. Sloane, Apr 10 2001

Conjecture: n is a sum of three cubes iff n is in this sequence.
As of their 2009 paper, Elsenhans and Jahnel did not know of a sum of three cubes that gives 33 or 42.
The problem with 33 is cracked, see links below: 8866128975287528^3 + (-8778405442862239)^3 + (-2736111468807040)^3 = 33. - Alois P. Heinz, Mar 11 2019
Numbers that are congruent to {0, 1, 2, 3, 6, 7, 8} mod 9. - Wesley Ivan Hurt, Jul 21 2016
Heath-Brown conjectures that n is a sum of three cubes in infinitely many ways iff n is in this sequence (and not at all otherwise). See his paper for a conjectural asymptotic. - Charles R Greathouse IV, Mar 12 2019
The problem with 42 is cracked by Andrew Booker from University of Bristol and Andrew Sutherland from Massachusetts Institute of Technology, see the link below: 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3. - Jianing Song, Sep 07 2019
A third solution to writing 3 as a sum of three third powers was found by the same team using 4 million computer-hours. 3 = 569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3. - Peter Luschny, Sep 20 2019

			30 belongs to this sequence because it has the partition as sum of 3 cubes 30 = (-283059965)^3 + (-2218888517)^3 + (2220422932)^3. - _Artur Jasinski_, Apr 30 2010, edited by _M. F. Hasler_, Nov 10 2015

R. K. Guy, Unsolved Problems in Number Theory, Section D5.
Cohen H. 2007. Number Theory Volume I: Tools and Diophantine Equations. Springer Verlag p. 380. - Artur Jasinski, Apr 30 2010

Harry J. Smith, Table of n, a(n) for n = 1..2000
Nikos Bagis, On the numbers that are sums of three cubes, arXiv:2009.11972 [math.GM], 2020.
Andrew R. Booker, Cracking the problem with 33, March 2019
Andrew R. Booker and Brady Haran, 42 is the new 33, Numberphile video (2019)
Andrew R. Booker and Brady Haran, NEWS: The Mystery of 42 is Solved, Numberphile video (2019)
Tim Browning and Brady Haran, The Uncracked Problem with 33, Numberphile video (2015)
Tim Browning and Brady Haran, 74 is cracked, Numberphile video (2016)
Jean-Louis Colliot-Thélène and Olivier Wittenberg, Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, Amer. J. Math. 134:5 (2012), pp. 1303-1327.
Andreas-Stephan Elsenhans and Jörg Jahnel, List of solutions of x^3 + y^3 + z^3 = n for n < 1000 neither a cube nor twice a cube
A.-S. Elsenhans, J. Jahnel, New sums of three cubes, Math. Comp. 78 (2009) 1227-1230.
Brady Haran, 569936821221962380720, Numberphile video (2020)
D. R. Heath-Brown, The density of zeros of forms for which weak approximation fails, Mathematics of Computation 59 (1992), pp. 613-623.
Sander G. Huisman, Newer sums of three cubes, arXiv:1604.07746 [math.NT], 2016.
H. Mishima, About n=x^3+y^3+z^3
Andrew Sutherland, Sums of three cubes, Slides of a talk given May 07 2020 on the Number Theory Web.
University of Bristol, Sum of three cubes for 42 finally solved - using real life planetary computer
Wikipedia, Manin obstruction
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A060465, A060466, A060467, A334521, A334522.

A156638 is the complement of this sequence.

A060464:=Filtered([0..100],n->n mod 9 <>4 and n mod 9 <>5); # Muniru A Asiru, Feb 17 2018

[n : n in [0..150] | n mod 9 in [0, 1, 2, 3, 6, 7, 8]]; // Wesley Ivan Hurt, Jul 21 2016

for n from 0 to 100 do if n mod 9 <> 4 and n mod 9 <> 5 then printf(`%d,`, n) fi:od:

a = {}; Do[If[(Mod[n, 9] == 4) || (Mod[n, 9] == 5), , AppendTo[a, n]], {n, 1, 300}]; a (* Artur Jasinski, Apr 30 2010 *)
Which[Mod[#,9]==4,Nothing,Mod[#,9]==5,Nothing,True,#]&/@Range[0,100] (* Harvey P. Dale, Jul 31 2023 *)

n=-1; for (m=0, 4000, if (m%9!=4 && m%9!=5, write("b060464.txt", n++, " ", m)); if (n==2000, break)) \\ Harry J. Smith, Jul 05 2009

concat(0, Vec(x^2*(x^3+x^2+1)*(x^3+x+1)/((1+x+x^2+x^3+x^4+x^5+x^6)*(x-1)^2) + O(x^100))) \\ Altug Alkan, Nov 06 2015

a(n)=n\7*9+[0, 1, 2, 3, 6, 7, 8][n%7+1] \\ Charles R Greathouse IV, Nov 06 2015

G.f.: x^2*(x^3+x^2+1)*(x^3+x+1) / ( (1+x+x^2+x^3+x^4+x^5+x^6)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jul 21 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8; a(n) = a(n-7) + 9 for n>7.
a(n) = (63*n - 63 + 2*(n mod 7) + 2*((n+1) mod 7) - 12*((n+2) mod 7) + 2*((n+3) mod 7) + 2*((n+4) mod 7) + 2*((n+5) mod 7) + 2*((n+6) mod 7))/49.
a(7k) = 9k-1, a(7k-1) = 9k-2, a(7k-2) = 9k-3, a(7k-3) = 9k-6, a(7k-4) = 9k-7, a(7k-5) = 9k-8, a(7k-6) = 9k-9. (End)

More terms from James Sellers, Apr 11 2001

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|.

Original entry on oeis.org

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

A060466 Value of y 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|.

Original entry on oeis.org

0, 0, 1, 1, -1, -1, 0, 1, 1, -2, 10, 2, -1609, 2, -2, -2, -2, -14, -15550555555, -1, -1, 0, 1, 1, -2218888517, -8778405442862239, 2, 2, 2, -3, -3, 134476, 80435758145817515, 2, -7, -3, 3, 7, -26, 659, 60702901317, 3, -11, 3, -21, -2, -4, -4, 3, -1, 0, 1, 1, -4, 20, 2, 9, 2

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) = -2218888517. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the second column of the table on Hisanori Mishima's web page. - M. F. Hasler, Nov 10 2015

			For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -1609.
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, Section D5.

A. Bogomolny, Finicky Diophantine Equations on cut-the-knot.org, accessed Nov. 10, 2015.
A.-S. Elsenhans, J. Jahnel, New sums of three cubes, Math. Comp. 78 (2009) 1227-1230.
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], 2009-2014.

Cf. A060465, A060467, A173515.

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]; A060466 = Table[y /. ToRules[ Simplify[ r[A060464[[k]], A060465[[k]]] /. C[1] -> 0]], {k, 1, Length[A060464]}] (* Jean-François Alcover, Jul 11 2012 *)

In order to be consistent with A060465, where only primitive solutions are selected, a(18)=2 was replaced with -15550555555, by Jean-François 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

A332201 Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| >= |y| >= |z|, or 0 if no such x exists.

Original entry on oeis.org

0, 1, 1, 1, 0, 0, 2, 2, 2, 2, 2, 3, -11, 0, 0, 2, 2, 2, 3, 3, 3, 16, 0, 0, 2, 3, 3, 3, 3, 3, 2220422932, 0, 0

Offset: 0

M. F. Hasler, Feb 08 2020

It is known that there is no solution for n congruent to +-4 (mod 9), but it is now conjectured that there is a solution (and probably infinitely many such) for all other numbers. The numbers n = 0, 1 and 2 are the only cases for which infinite families of parametric solutions are known, for other n the solutions seem to be sporadic.
Search on this problem was motivated by a statement in Mordell's paper from 1953. Beck et al. found a solution for n = 30 in 1999, and for 52 in 2000. Huisman found a solution for n = 74 in 2016. A solution for 33 was found by Booker in 2019. The number 42 was the last one below 100 for which a solution was found, in late 2019, using a collaborative effort with supercomputers and home computers from volunteers.
For n < 30, we have a(n) = A246869(n+1) for the nonzero values, while A246869(n+1) = 2 for n == 4 or 5 (mod 9) up to there.

			   0 = 0^3 + 0^3 + 0^3,     1 = 1^3 + 0^3 + 0^3,
   2 = 1^3 + 1^3 + 0^3,     3 = 1^3 + 1^3 + 1^3,
   6 = 2^3 - 1^3 - 1^3,     7 = 2^3 - 1^3 + 0^3,
   8 = 2^3 + 0^3 + 0^3,     9 = 2^3 + 1^3 + 0^3,
  10 = 2^3 + 1^3 + 1^3,    11 = 3^3 - 2^3 - 2^3,
  12 = -11^3 + 10^3 + 7^3, 15 = 2^3 + 2^3 - 1^3,
  16 = 2^3 + 2^3 + 0^3,    17 = 2^3 + 2^3 + 1^3,
  18 = 3^3 - 2^3 - 1^3,    19 = 3^3 - 2^3 + 0^3,
  20 = 3^3 - 2^3 + 1^3,    21 = 16^3 - 14^3 - 11^3,
  24 = 2^3 + 2^3 + 2^3,    25 = 3^3 - 1^3 - 1^3,
  26 = 3^3 - 1^3 + 0^3,    27 = 3^3 + 0^3 + 0^3,
  28 = 3^3 + 1^3 + 0^3,    29 = 3^3 + 1^3 + 1^3,
30 = 2220422932^3 - 2218888517^3 - 283059965^3 was discovered by Beck, Pine, Yarbrough and Tarrant in 1999 following an approach suggested by N. Elkies.
33 = 8866128975287528^3 - 8778405442862239^3 - 2736111468807040^3 was found by A. Booker in 2019. It is uncertain whether these are the smallest solutions.

Michael Beck, Eric Pine, Wayne Tarrant and Kim Yarbrough Jensen, New integer representations as the sum of three cubes, Math. Comp. 76 (2007) pp. 1683-1690, DOI:10.1090/S0025-5718-07-01947-3.
A. R. Booker, Cracking the problem with 33, Res. Number Theory 5 no. 26 (2019), DOI:10.1007/s40993-019-0162-1.
V. L. Gardiner, R. B. Lazarus and P. R. Stein: Solutions of the Diophantine Equation x^3 + y^3 = z^3 - d, Math.Comp. 18, No. 87 (1964), pp. 408-413. DOI: 10.2307/2003763.
Jon Grantham and P. G. Walsh, Representing integers as a sum of three cubes, arXiv preprint, arXiv:2211.12149 [math.NT], 2022.
B. Haran, Sum of Three Cubes - Numberphile, YouTube playlist (featuring videos from Nov 06 2015, May 31 2016 (about 74), Mar 12 2019 (about 33), Sep 2019 (about 42)).
Sander G. Huisman, Newer sums of three cubes, arXiv:1604.07746 [math.NT] (2016).
J. C. P. Miller and M. F. C. Woollett, Solutions of the Diophantine Equation x^3+y^3+z^3=k, Journal of the London Mathematical Society, s1-30 (1955) pp. 101-110. DOI: 10.1112/jlms/s1-30.1.101.
L. J. Mordell, Integer Solutions of the Equation x^2+y^2+z^2+2xyz = n, Journal London Math. Soc. s1-28 no. 4 (1953) pp. 500-510.
Bjorn Poonen, Undecidability in number theory, Notices Amer. Math. Soc. 55 (2008), no. 3, pp. 344-350.

Cf. A060464, A060465, A060466, A060467, A246869.

apply( A332201(n,L=oo)={!bittest(48,n%9)&& for(c=0,L, my(t1=c^3-n, t2=c^3+n, a); for(b=0,c,((ispower(t1-b^3,3,&a)&&abs(a)<=c)||(ispower(t1+b^3,3,&a)&&abs(a)<=c))&&return(c); ispower(t2-b^3,3,&a) && abs(a)<=c && return(-c)))}, [0..29])

a(n) = 0 for n == 4 or n == 5 (mod 9).
a(n) <= k if |n - k^3| < 3 or |n - 2*k^3| < 2 or n = 3*k^3 for some k.
a(n) = A246869(n+1) for all n < 30 with a(n) > 0.

a(31) = a(32) = 0 added by Jinyuan Wang, Feb 15 2020

cp's OEIS Frontend

A060464 Numbers that are not congruent to 4 or 5 mod 9.

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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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A060466 Value of y 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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A332201 Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| >= |y| >= |z|, or 0 if no such x exists.

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