cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A159843 Sums of two rational cubes.

Original entry on oeis.org

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
Offset: 1

Views

Author

Steven Finch, Apr 23 2009

Keywords

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

Crossrefs

Complement of A185345.
Subsequences include A045980, A004999, and A003325.
Cf. A020894, A020895, A020897, A020898.

Programs

  • Mathematica
    (* 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 *)
  • 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

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

A293645 Positive numbers that are the sum of two (possibly negative) coprime cubes.

Original entry on oeis.org

1, 2, 7, 9, 19, 26, 28, 35, 37, 61, 63, 65, 91, 98, 117, 124, 126, 127, 133, 152, 169, 189, 215, 217, 218, 271, 279, 316, 331, 335, 341, 342, 344, 351, 370, 386, 387, 397, 407, 468, 469, 485, 511, 513, 539, 547, 559, 602, 604, 631, 637, 657, 665, 721, 728, 730
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Also sum or difference of two coprime cubes. - David A. Corneth, Oct 20 2017

Examples

			19 = 3^3 + (-2)^3, where 3 and -2 are coprime, so 19 is in the sequence.
152 = 5^3 + 3^3, where 5 and 3 are coprime, so 152 is in the sequence.

Links

Crossrefs

Cf. A003325 (positive cubes); A020895 (cubefree); A293646 (only coprime); A293647, A293650.

Programs

  • Maple
    filter:= proc(n) local s,x,y;
  for s in numtheory:-divisors(n) do
    x:= s/2 + sqrt(12*n/s-3*s^2)/6;
    if not x::integer then next fi;
    y:= s - x;
    if igcd(x,y) = 1 then return true fi;
  od;
  false
end proc:
select(filter, [seq(seq(9*i+j,j=[1,2,7,8,9]),i=0..1000)]); # Robert Israel, Oct 22 2017
  • Mathematica
    smax = 100000; (* upper limit for last term *)
m0 = smax^(1/3) // Ceiling;
f[m_] := f[m] = Module[{c, s, d}, Table[c = CoprimeQ[i^3, j^3]; {s = i^3 + j^3; If[0 < s <= smax && c, s, Nothing], d = j^3 - i^3; If[0 < d <= smax && c, d, Nothing]}, {i, 0, m}, {j, i, m}] // Flatten // Union];
f[m = m0];
f[m += m0];
While[f[m] != f[m - m0], m += m0];
f[m] (* Jean-François Alcover, Jun 28 2023 *)
  • PARI
    upto(lim) = {my(res = List([2]), c, i, j); for(i=1,sqrtnint(lim, 3), for(j=0, sqrtnint(lim - i^3, 3), if(gcd(i, j) == 1, listput(res, c)))); for(i=1, sqrtint(lim\3)+1, for(j = 1, i, if(gcd(i, j) == 1, c = i^3 - (i-j)^3; if(c<=lim, listput(res, c), next(2))))); listsort(res, 1); res} \\ David A. Corneth, Oct 20 2017

A293646 Sum of two (possibly negative) coprime cubes, but not the sum of 2 non-coprime cubes.

Original entry on oeis.org

1, 2, 7, 9, 19, 26, 28, 35, 37, 61, 63, 65, 91, 98, 117, 124, 126, 127, 133, 169, 215, 217, 218, 271, 279, 316, 331, 335, 341, 342, 344, 351, 370, 386, 387, 397, 407, 468, 469, 485, 511, 539, 547, 559, 602, 604, 631, 637, 657, 665, 721, 730, 737, 793, 817, 819
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Not every term is cubefree; some are sb^3 where s is in A159843 and b > 1.

Examples

			344 = 7^3 + 1^3 and 344 is not also the sum of cubes of 2 non-coprime integers, so 344 is in the sequence.
152 = 6^3 + (-4)^3 and 6 and -4 are not coprime, so 152 is not in the sequence.

Links

Crossrefs

Cf. A020895 (cubefree); A293645 (allows non-coprime); A293648, A293651

Programs

  • Mathematica
    s[n_] := CoprimeQ @@@ ({x, y} /. Solve[n == x^3 + y^3, {x, y}, Integers]);
Reap[For[k = 1, k < 2000, k++, If[Union[s[k]] == {True}, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Feb 02 2023 *)
Showing 1-3 of 3 results.

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