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

2, 5, 25, 50, 125, 625, 1250, 65, 15625, 31250, 78125, 390625, 781250, 325, 9765625, 19531250, 48828125, 244140625, 488281250, 1625, 6103515625, 12207031250, 30517578125, 4225, 8450, 8125, 3814697265625, 7629394531250, 19073486328125, 95367431640625

Offset: 1

Author

Altug Alkan, May 18 2016

Keywords

• nonn

Examples

			a(1) = 2 because 2^3 = 2^2 + 2^2.
a(2) = 5 because 5^3 = 5^2 + 10^2 = 2^2 + 11^2.
a(3) = 25 because 25^3 = 35^2 + 120^2 = 44^2 + 117^2 = 75^2 + 100^2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A006339, A016032, A025426, A084888.

Programs

Mathematica

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 *)
(* 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 *)

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
a(n)=my(k=1); while(A025426(k++^3)!=n, ); k
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

Extensions

a(10)-a(30) from Giovanni Resta, May 18 2016