A185345 Numbers that are not the sum of two rational cubes.
3, 4, 5, 10, 11, 14, 18, 21, 23, 24, 25, 29, 32, 36, 38, 39, 40, 41, 44, 45, 46, 47, 52, 55, 57, 59, 60, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, 101, 102, 108, 109, 111, 112, 113, 116, 118, 119, 121, 122, 129, 131, 135, 137, 138, 144, 145, 146, 147
Offset: 1
Keywords
Examples
22 can be expressed as (17299/9954)^3 + (25469/9954)^3, so 22 is not in the sequence.
References
- Henri Cohen, Number Theory - Volume I: Tools and Diophantine Equations, Springer-Verlag, 2007, pp. 378-379.
- Yu. I. Manin, A. A. Panchishkin, Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Second Edition), Springer-Verlag, 2006, pp. 43-46.
Links
- Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
- H. Nakao, [2017.11.14] Rational Points on Elliptic Curves: x^3+y^3=n (n in [1..10000]). This sequence consists of the numbers that have a dash in the corresponding cell of the next-to-last column of the table.
- Ernst S. Selmer, The diophantine equation ax^3 + by^3 + cz^3 = 0, Acta Math. 85 (1951), pp. 203-362.
- Index entries for sequences related to sums of cubes
Crossrefs
Programs
-
Magma
lst1:=[]; lst2:=[x^3+y^3: x, y in [0..5]]; for n in [1..147] do if IsZero(Rank(EllipticCurve([0, 16*n^2]))) and not n in lst2 then lst1:=Append(lst1, n); end if; end for; lst1;
-
Mathematica
(* A naive program with a few pre-computed terms from A159843 *) nmax = 122; xmax = 3000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]}& /@ FactorInteger[n]); nn = Join[{1}, Reap[Do[n = CubeFreePart[x*y*(x+y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union]; A159843 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 89, 94, 103, 107, 122}], # <= nmax &]; Complement[Range[nmax], A159843] (* Jean-François Alcover, Feb 10 2015 *)
-
PARI
isok(k) = my(v=thue('x^3+1, k)); if(!(#v>0 && #select(k->k>=0, concat(v))>#v) && ellanalyticrank(ellinit([0, 16*k^2]))[1]==0, 1, 0); \\ Arkadiusz Wesolowski, May 21 2023