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A185345
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Numbers that are not the sum of two rational cubes.
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3
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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
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OFFSET
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1,1
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REFERENCES
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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.
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LINKS
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EXAMPLE
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22 can be expressed as (17299/9954)^3 + (25469/9954)^3, so 22 is not in the sequence.
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MATHEMATICA
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(* 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 *)
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PROG
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(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
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CROSSREFS
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Complement of A159843. One subsequence of this sequence is A022555, numbers that are not the sum of two nonnegative integer cubes.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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