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 A159843 Sums of two rational cubes. 20
 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, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378. The prime elements are listed in A166246. - Max Alekseyev, Oct 10 2009 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 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 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 REFERENCES H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 379. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 Levent Alpöge, Manjul Bhargava, and Ari Shnidman, Integers expressible as the sum of two rational cubes, arXiv:2210.10730 [math.NT], Oct. 2022. Somnath Jha, Dipramit Majumdar, and B. Sury, Infinitely many primes in each of the residue classes \$1\$ and \$8\$ modulo \$9\$ are sums of two rational cubes, arXiv preprint (2023). arXiv:2301.06970 [math.NT] Index entries for sequences related to sums of cubes FORMULA 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 MATHEMATICA (* A naive program with a few pre-computed terms *) nmax = 117; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[], Mod[#[], 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 *) PROG (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, return(1)); if(mwr<1, return(0)); ar=ellanalyticrank(E); if(ar<2, return(ar)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr>0, return(1), mwr<1, return(0))); "yes under BSD conjecture" \\ Charles R Greathouse IV, Dec 02 2022 CROSSREFS Complement of A185345. Subsequences include A045980, A004999, and A003325. Cf. A020894, A020895, A020897, A020898. Sequence in context: A047555 A184939 A043050 * A243652 A080780 A138168 Adjacent sequences: A159840 A159841 A159842 * A159844 A159845 A159846 KEYWORD nice,nonn AUTHOR Steven Finch, Apr 23 2009 STATUS approved

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Last modified June 6 05:06 EDT 2023. Contains 363139 sequences. (Running on oeis4.)