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A273238 Least number k such that k^3 is the sum of two nonzero squares in exactly n ways.

%I A273238 #19 May 19 2016 17:09:21
%S A273238 2,5,25,50,125,625,1250,65,15625,31250,78125,390625,781250,325,
%T A273238 9765625,19531250,48828125,244140625,488281250,1625,6103515625,
%U A273238 12207031250,30517578125,4225,8450,8125,3814697265625,7629394531250,19073486328125,95367431640625
%N A273238 Least number k such that k^3 is the sum of two nonzero squares in exactly n ways.
%H A273238 Giovanni Resta, <a href="/A273238/b273238.txt">Table of n, a(n) for n = 1..100</a>
%e A273238 a(1) = 2 because 2^3 = 2^2 + 2^2.
%e A273238 a(2) = 5 because 5^3 = 5^2 + 10^2 = 2^2 + 11^2.
%e A273238 a(3) = 25 because 25^3 = 35^2 + 120^2 = 44^2 + 117^2 = 75^2 + 100^2.
%t A273238 Function[t, FirstPosition[t, #] & /@ Range@ 8]@ Map[Length@ Select[ PowersRepresentations[#^3, 2, 2], ! MemberQ[#, 0] &] &, Range[2 10^3]] // Flatten (* _Michael De Vlieger_, May 18 2016 *)
%t A273238 (* code for first 100 terms *) nR[n_] := (SquaresR[2, n] + Plus @@ Pick[{-4, 4}, IntegerQ /@ Sqrt[{n, n/2}]])/8; c[w_] := Floor[1/2 Times @@ (3 w + 1)]; q[1] = 2; q[n_] := Min[Reap[Do[ x = Times @@ (Take[{5, 13, 17, 29}, Length[e]]^e); If[c[e] == n && nR[x^3] == n, Sow[x]]; If[c[e] + 1 == n && nR[8 x^3] == n, Sow[2 x]], {e, Join[Transpose[{ Range@ 80}], Join @@ (IntegerPartitions[#, 4] & /@ Range[21]) ]}]][[2, 1]]]; Array[q, 100] (* _Giovanni Resta_, May 18 2016 *)
%o A273238 (PARI) A025426(n)=my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2
%o A273238 a(n)=my(k=1); while(A025426(k++^3)!=n, ); k
%o A273238 first(n)=my(v=vector(n),t,k); while(1, t=A025426(k++^3); if(t>0 && t<=n && v[t]==0, v[t]=k; if(factorback(v), return(v)))) \\ _Charles R Greathouse IV_, May 18 2016
%Y A273238 Cf. A006339, A016032, A025426, A084888.
%K A273238 nonn
%O A273238 1,1
%A A273238 _Altug Alkan_, May 18 2016
%E A273238 a(10)-a(30) from _Giovanni Resta_, May 18 2016