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%I #35 Mar 18 2023 08:49:14
%S 1,2,6,7,8,9,12,13,15,16,17,19,20,22,26,27,28,30,31,33,34,35,37,42,43,
%T 48,49,50,51,53,54,56,58,61,62,63,64,65,67,68,69,70,71,72,75,78,79,84,
%U 85,86,87,89,90,91,92,94,96,97,98,103,104,105,106,107,110,114,115,117
%N Sums of two rational cubes.
%C Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378.
%C The prime elements are listed in A166246. - _Max Alekseyev_, Oct 10 2009
%C Alpöge et al. prove 'that the density of integers expressible as the sum of two rational cubes is strictly positive and strictly less than 1.' The authors remark that it is natural to conjecture that these integers 'have natural density exactly 1/2.' - _Peter Luschny_, Nov 30 2022
%C Jha, Majumdar, & Sury prove that every nonzero residue class mod p (for prime p) has infinitely many elements, as do 1 and 8 mod 9. - _Charles R Greathouse IV_, Jan 24 2023
%C Alpöge, Bhargava, & Shnidman prove that the lower density of this sequence is at least 2/21 and its upper density is at most 5/6. - _Charles R Greathouse IV_, Feb 15 2023
%D H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 379.
%H Charles R Greathouse IV, <a href="/A159843/b159843.txt">Table of n, a(n) for n = 1..10000</a>
%H Levent Alpöge, Manjul Bhargava, and Ari Shnidman, <a href="https://arxiv.org/abs/2210.10730">Integers expressible as the sum of two rational cubes</a>, arXiv:2210.10730 [math.NT], Oct. 2022.
%H Somnath Jha, Dipramit Majumdar, and B. Sury, <a href="https://arxiv.org/abs/2301.06970">Infinitely many primes in each of the residue classes $1$ and $8$ modulo $9$ are sums of two rational cubes</a>, arXiv preprint (2023). arXiv:2301.06970 [math.NT]
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A cubefree integer c>2 is in this sequence iff the elliptic curve y^2=x^3+16*c^2 has positive rank. - _Max Alekseyev_, Oct 10 2009
%t (* A naive program with a few pre-computed terms *) nmax = 117; xmax = 2000; 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, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* _Jean-François Alcover_, Apr 03 2012 *)
%o (PARI) is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<3 || mwr[1], return(1)); if(mwr[2]<1, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(ar)); for(effort=1,99, mwr=ellrank(eri,effort); if(mwr[1]>0, return(1), mwr[2]<1, return(0))); "yes under BSD conjecture" \\ _Charles R Greathouse IV_, Dec 02 2022
%Y Complement of A185345.
%Y Subsequences include A045980, A004999, and A003325.
%Y Cf. A020894, A020895, A020897, A020898.
%K nice,nonn
%O 1,2
%A _Steven Finch_, Apr 23 2009
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