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Least number of the form x^n + y^n that is the sum of two positive squares in exactly n ways (x, y > 0).
0

%I #11 Jun 27 2016 09:53:20

%S 2,50,3925,50881,625000000,117650,1562500000000,815737282,

%T 3906250000000000,25937425625,9765625000000000000,23298085126577,

%U 24414062500000000000000,7862651120078217664621465531250,61035156250000000000000000,10000000000000001,152587890625000000000000000000

%N Least number of the form x^n + y^n that is the sum of two positive squares in exactly n ways (x, y > 0).

%e a(2) = 50 because 50 = 1^2 + 7^2 = 5^2 + 5^2.

%e a(3) = 3925 because 3925 = 12^3 + 13^3 = 9^2 + 62^2 = 26^2 + 57^2 = 30^2 + 55^2.

%e a(4) = 50881 because 50881 = 4^4 + 15^4 = 16^2 + 225^2 = 65^2 + 216^2 = 120^2 + 191^2 = 159^2 + 160^2.

%t nR[n_]:= (SquaresR[2, n]+ Plus@@ Pick[{-4, 4}, IntegerQ/@ Sqrt[{n, n/2} ]])/8; a[1]=2; a[n_]:= Block[{m = 10^3, s, z}, While[{} == (s = Reap[ Do[ If[nR[z = x^n + y^n] == n, Sow@ z], {x, Floor[m^(1/n)]}, {y, Min[x, Floor[m - x^n]^(1/n)]}]][[2]]), m *= 2]; Min[s[[1]]]]; Array[a, 10] (* _Giovanni Resta_, Jun 27 2016 *)

%K nonn

%O 1,1

%A _Altug Alkan_, Jun 24 2016

%E a(5)-a(17) from _Giovanni Resta_, Jun 27 2016