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%I #43 Mar 16 2016 07:45:31
%S 2,9,28,35,65,91,126,133,152,189,217,341,344,351,370,407,468,513,539,
%T 559,637,730,737,793,854,855,1001,1027,1072,1241,1332,1339,1343,1358,
%U 1395,1456,1547,1674,1729,1843,1853,2060,2071,2198,2205,2224,2261,2322,2331,2413
%N Numbers that are sums of two coprime positive cubes.
%C Not a subsequence of A020898: non-cubefree members of this sequence include 152, 189, 344, 351, 513, 1072. - _Robert Israel_, Mar 16 2016
%H Arkadiusz Wesolowski, <a href="/A202679/b202679.txt">Table of n, a(n) for n = 1..10000</a>
%H R. C. Baker, <a href="http://dx.doi.org/10.4064/aa129-2-1">Sums of two relatively prime cubes</a>, Acta Arithmetica 129(2007), 103-146.
%H Kevin A. Broughan, <a href="http://www.math.waikato.ac.nz/~kab/papers/cubic4.pdf">A computational approach to characterizing the sum of two cubes</a>, Hamilton: University of Waikato, 2001, p. 9.
%H P. Erdős and K. Mahler, <a href="http://carmaweb.newcastle.edu.au/mahler/docs/047.pdf">On the number of integers which can be represented by a binary form</a>, J. London Math. Soc. 13 (1938), pp. 134-139. [<a href="http://www.renyi.hu/~p_erdos/1938-10.pdf">alternate link</a>]
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1939-10.pdf">On the integers of the form x^k + y^k</a>, J. London Math. Soc. 14 (1939), pp. 250-254.
%H <a href="/index/Su#ssq">Index to sequences related to sums of cubes</a>
%F 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
%e 28 is in the sequence since 1^3 + 3^3 = 28 and (1, 3) = 1.
%p N:= 10000: # to get all terms <= N
%p 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)))}:
%p sort(convert(S,list)); # _Robert Israel_, Mar 15 2016
%t 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)}]]]
%t Select[Range@ 2500, Length[PowersRepresentations[#, 2, 3] /. {{0, _} -> Nothing, {a_, b_} /; ! CoprimeQ[a, b] -> Nothing}] > 0 &] (* _Michael De Vlieger_, Mar 15 2016 *)
%o (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
%o (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
%Y Subsequence of A003325.
%Y Cf. A024670, A001235, A018850.
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Jan 06 2012
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