1,1

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.

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.

22 can be expressed as (17299/9954)^3 + (25469/9954)^3, so 22 is not in the sequence.

(* 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 *)

(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;

(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

Complement of A159843. One subsequence of this sequence is A022555, numbers that are not the sum of two nonnegative integer cubes.

Sequence in context: A047364 A274519 A139445 * A260823 A359486 A135114

Adjacent sequences: A185342 A185343 A185344 * A185346 A185347 A185348

Arkadiusz Wesolowski, Mar 17 2012

approved