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%I #40 Feb 04 2016 20:58:19
%S 9,35,54,65,72,91,126,133,152,189,217,224,243,250,280,341,344,370,432,
%T 468,513,539,576,637,686,728,730,737,756,793,854,945,1001,1027,1064,
%U 1072,1125,1216,1241,1332,1339,1358,1395,1456,1458,1512,1547,1674,1729,1736,1755,1843,1853
%N Numbers that are the sum of 3 nonzero squares (A000408) and the sum of 2 positive cubes (A003325).
%C Intersection of A000408 and A003325.
%C Sequence focuses on the solutions of equation x^3 + y^3 = a^2 + b^2 + c^2 where x, y, a, b, c > 0.
%H Robert Israel, <a href="/A267702/b267702.txt">Table of n, a(n) for n = 1..3649</a>
%e 9 is a term because 9 = 1^3 + 2^3 = 1^2 + 2^2 + 2^2.
%e 35 is a term because 35 = 2^3 + 3^3 = 1^2 + 3^2 + 5^2.
%e 54 is a term because 54 = 3^3 + 3^3 = 3^2 + 3^2 + 6^2.
%p N:= 1000: # to get all terms <= N
%p S3:= {seq(seq(seq(a^2+b^2+c^2, c = b .. floor(sqrt(N-a^2-b^2))),
%p b=a .. floor(sqrt((N-a^2)/2))), a = 1 .. floor(sqrt(N/3)))}:
%p C2:= {seq(seq(a^3+b^3, b = a .. floor((N-a^3)^(1/3))),a = 1 .. floor((N/2)^(1/3)))}:
%p sort(convert(S3 intersect C2, list)); # _Robert Israel_, Jan 25 2016
%o (PARI) isA000408(n) = {my(a, b); a=1; while(a^2+1<n, b=1; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1)); b++; ); a++; ); return(0); }
%o T=thueinit('z^3+1);
%o isA003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0;
%o for(n=3, 1e4, if(isA000408(n) && isA003325(n), print1(n, ", ")));
%Y Cf. A000408, A003325.
%K nonn
%O 1,1
%A _Altug Alkan_, Jan 23 2016
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