A202679 Numbers that are sums of two coprime positive cubes.

2, 9, 28, 35, 65, 91, 126, 133, 152, 189, 217, 341, 344, 351, 370, 407, 468, 513, 539, 559, 637, 730, 737, 793, 854, 855, 1001, 1027, 1072, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1547, 1674, 1729, 1843, 1853, 2060, 2071, 2198, 2205, 2224, 2261, 2322, 2331, 2413

Offset

1,1

Comments

Not a subsequence of A020898: non-cubefree members of this sequence include 152, 189, 344, 351, 513, 1072. - Robert Israel, Mar 16 2016

Links

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

R. C. Baker, Sums of two relatively prime cubes, Acta Arithmetica 129(2007), 103-146.

Kevin A. Broughan, A computational approach to characterizing the sum of two cubes, Hamilton: University of Waikato, 2001, p. 9.

P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), pp. 134-139. [alternate link]

P. Erdős, On the integers of the form x^k + y^k, J. London Math. Soc. 14 (1939), pp. 250-254.

Index to sequences related to sums of cubes

Formula

Erdős & Mahler shows that a(n) < kn^(3/2) for some k. Erdős later gives an elementary proof. - Charles R Greathouse IV, Dec 05 2012

Example

28 is in the sequence since 1^3 + 3^3 = 28 and (1, 3) = 1.

Maple

N:= 10000: # to get all terms <= N
S:= {2, seq(seq(x^3 + y^3, y = select(t -> igcd(t, x)=1, [$x+1 .. floor((N - x^3)^(1/3))])), x = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S, list)); # Robert Israel, Mar 15 2016

Mathematica

nn = 2500; Union[Flatten[Table[If[CoprimeQ[x, y] == True, x^3 + y^3, {}], {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]
Select[Range@ 2500, Length[PowersRepresentations[#, 2, 3] /. {{0, _} -> Nothing, {a_, b_} /; ! CoprimeQ[a, b] -> Nothing}] > 0 &] (* Michael De Vlieger, Mar 15 2016 *)

Prog

(PARI) is(n)=for(k=1, (n\2+.5)^(1/3), if(gcd(k, n)==1&&ispower(n-k^3, 3), return(1))); 0 \\ Charles R Greathouse IV, Apr 13 2012
(PARI) list(lim)=my(v=List()); forstep(x=1, lim^(1/3), 2, forstep(y=2, (lim-x^3+.5)^(1/3), 2, if(gcd(x, y)==1, listput(v, x^3+y^3))); forstep(y=1, min((lim-x^3+.5)^(1/3), x), 2, if(gcd(x, y)==1, listput(v, x^3+y^3)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Dec 05 2012

Crossrefs

Subsequence of A003325.

Cf. A024670, A001235, A018850.

Sequence in context: A357721 A155472 A100293 * A340049 A256467 A303373

Adjacent sequences: A202676 A202677 A202678 * A202680 A202681 A202682

Keyword

nonn

Author

Arkadiusz Wesolowski, Jan 06 2012

Status

approved