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
Keywords
Examples
28 is in the sequence since 1^3 + 3^3 = 28 and (1, 3) = 1.
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
Programs
-
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 *) -
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
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
Comments