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%I #15 May 22 2016 00:14:19
%S 8,125,3125,4225,1953125,48828125,105625,274625,762939453125,2640625,
%T 476837158203125,17850625,1221025,34328125,186264514923095703125,
%U 1650390625,446265625,1160290625,41259765625,4291015625,45474735088646411895751953125,30525625
%N Least perfect power that is the sum of two nonzero squares in exactly n ways.
%C Least m^k that is the sum of two nonzero squares in exactly n ways where m > 0 and k >= 2.
%C Terms of this sequence are 2^3, 5^3, 5^5, 65^2, 5^10, 5^11, 325^2, 65^3, ...
%C Prime powers that are listed in this sequence are 2^3, 5^3, 5^5, 5^10, 5^11, ...
%H Giovanni Resta, <a href="/A273279/b273279.txt">Table of n, a(n) for n = 1..100</a>
%e 8 is a term because 8 = 2^3 = 2^2 + 2^2.
%e 125 is a term because 125 = 5^3 = 2^2 + 11^2 = 5^2 + 10^2.
%e 3125 is a term because 3125 = 5^5 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2.
%t p = Select[Prime@ Range@ 90, Mod[#, 4] == 1 &]; f[w_] := Times @@ (Take[p, Length@w]^Reverse[w]); c[w_] := Floor[(1/2) Times @@ (w+1)];r[w_] := Block[{v, k = If[Length@w == 1, 1,2]}, While[(v = cn[k w]) < trg, k++]; If[v == trg, b = Min[b, f[k*w]]]; If[cn[w] < trg, r[Append[w, 1]]; v=w; v[[-1]]++; r[v]]]; a[1]=8; a[n_] := (b=Infinity; trg = n; r[{2}]; r[{1, 1}]; b); Array[a, 50] (* _Giovanni Resta_, May 19 2016 *)
%Y Cf. A001597, A006339, A016032, A025426, A266927, A273238.
%K nonn
%O 1,1
%A _Altug Alkan_, May 19 2016
%E a(9)-a(22) from _Giovanni Resta_, May 19 2016
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