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%I #11 Jun 06 2016 05:33:28
%S 1,8,14,27,36,63,64,76,112,140,172,185,216,234,260,288,343,364,378,
%T 427,504,512,536,608,666,679,728,729,868,896,972,1000,1030,1099,1112,
%U 1120,1161,1270,1331,1376,1404,1463,1480,1628,1688,1728,1750,1764,1859,2052,2080,2156
%N Numbers that are the average of 3 nonzero squares and the average of 2 positive cubes.
%C Values of (x^3 + y^3)/2 such that (x^3 + y^3)/2 = (a^2 + b^2 + c^2)/3 where x, y, a, b, c > 0, is soluble.
%e 14 is a term because 14 = (1^3 + 3^3)/2 = (1^2 + 4^2 + 5^2)/3.
%t repQ[n_,k_,e_] := {} != Quiet@ IntegerPartitions[n, {k}, Range[n^ (1/e) ]^e, 1]; Select[Range@ 2156, repQ[2*#,2,3] && repQ[3*#,3,2] &] (* _Giovanni Resta_, Jun 03 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 isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
%o lista(nn) = for(n=1, nn, if(isA003325(2*n) && isA000408(3*n), print1(n, ", ")));
%Y Cf. A000408, A003325, A267702.
%K nonn,easy
%O 1,2
%A _Altug Alkan_, Jun 01 2016
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