A185345 - OEIS

%I #43 May 24 2023 07:33:46

%S 3,4,5,10,11,14,18,21,23,24,25,29,32,36,38,39,40,41,44,45,46,47,52,55,

%T 57,59,60,66,73,74,76,77,80,81,82,83,88,93,95,99,100,101,102,108,109,

%U 111,112,113,116,118,119,121,122,129,131,135,137,138,144,145,146,147

%N Numbers that are not the sum of two rational cubes.

%D Henri Cohen, Number Theory - Volume I: Tools and Diophantine Equations, Springer-Verlag, 2007, pp. 378-379.

%D Yu. I. Manin, A. A. Panchishkin, Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Second Edition), Springer-Verlag, 2006, pp. 43-46.

%H Arkadiusz Wesolowski, <a href="/A185345/b185345.txt">Table of n, a(n) for n = 1..10000</a>

%H H. Nakao, <a href="http://www.kaynet.or.jp/~kay/misc/nna10000.html">[2017.11.14] Rational Points on Elliptic Curves: x^3+y^3=n (n in [1..10000])</a>. This sequence consists of the numbers that have a dash in the corresponding cell of the next-to-last column of the table.

%H Ernst S. Selmer, <a href="http://dx.doi.org/10.1007/BF02395746">The diophantine equation ax^3 + by^3 + cz^3 = 0</a>, Acta Math. 85 (1951), pp. 203-362.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>

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

%t (* 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 *)

%o (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;

%o (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

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

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Mar 17 2012