

A185345


Numbers that are not the sum of two rational cubes.


2



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
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OFFSET

1,1


REFERENCES

Henri Cohen, Number Theory  Volume I: Tools and Diophantine Equations, SpringerVerlag, 2007, pp. 378379.
Yu. I. Manin, A. A. Panchishkin, Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Second Edition), SpringerVerlag, 2006, pp. 4346.


LINKS

Table of n, a(n) for n=1..52.
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 nexttolast column of the table.
Ernst S. Selmer, The diophantine equation ax^3 + by^3 + cz^3 = 0, Acta Math. 85 (1951), pp. 203362.
Index entries for sequences related to sums of cubes


EXAMPLE

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


MATHEMATICA

(* A naive program with a few precomputed 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] (* JeanFrançois Alcover, Feb 10 2015 *)


PROG

(MAGMA) lst1:=[]; lst2:=[x^3+y^3: x, y in [0..3]]; for n in [1..40] do if IsZero(Rank(EllipticCurve([0, 16*n^2]))) and not n in lst2 then lst1:=Append(lst1, n); end if; end for; lst1;


CROSSREFS

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 A135114 A240789
Adjacent sequences: A185342 A185343 A185344 * A185346 A185347 A185348


KEYWORD

nonn,more


AUTHOR

Arkadiusz Wesolowski, Mar 17 2012


STATUS

approved



