This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A159843 #41 Sep 29 2024 12:28:44
%S A159843 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 A159843 48,49,50,51,53,54,56,58,61,62,63,64,65,67,68,69,70,71,72,75,78,79,84,
%U A159843 85,86,87,89,90,91,92,94,96,97,98,103,104,105,106,107,110,114,115,117
%N A159843 Sums of two rational cubes.
%C A159843 Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378.
%C A159843 The prime elements are listed in A166246. - _Max Alekseyev_, Oct 10 2009
%C A159843 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 A159843 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 A159843 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 A159843 H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 379.
%H A159843 Charles R Greathouse IV, <a href="/A159843/b159843.txt">Table of n, a(n) for n = 1..10000</a>
%H A159843 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 A159843 Bogdan Grechuk, <a href="https://doi.org/10.1007/978-3-031-62949-5_6">Existence of Non-Trivial Solutions to Homogeneous Equations</a>, Polynomial Diophantine Equations, Springer, Cham (2024), Chapter 6, 473-536.
%H A159843 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 arXiv:2301.06970 [math.NT], 2023-2024.
%H A159843 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A159843 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 A159843 (* 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 A159843 (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 A159843 Complement of A185345.
%Y A159843 Subsequences include A045980, A004999, and A003325.
%Y A159843 Cf. A020894, A020895, A020897, A020898.
%K A159843 nice,nonn
%O A159843 1,2
%A A159843 _Steven Finch_, Apr 23 2009