A273279 Least perfect power that is the sum of two nonzero squares in exactly n ways.

8, 125, 3125, 4225, 1953125, 48828125, 105625, 274625, 762939453125, 2640625, 476837158203125, 17850625, 1221025, 34328125, 186264514923095703125, 1650390625, 446265625, 1160290625, 41259765625, 4291015625, 45474735088646411895751953125, 30525625

Offset: 1

Author

Altug Alkan, May 19 2016

Keywords

• nonn

Comments

Least m^k that is the sum of two nonzero squares in exactly n ways where m > 0 and k >= 2.

Terms of this sequence are 2^3, 5^3, 5^5, 65^2, 5^10, 5^11, 325^2, 65^3, ...

Prime powers that are listed in this sequence are 2^3, 5^3, 5^5, 5^10, 5^11, ...

Examples

			8 is a term because 8 = 2^3 = 2^2 + 2^2.
125 is a term because 125 = 5^3 = 2^2 + 11^2 = 5^2 + 10^2.
3125 is a term because 3125 = 5^5 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2.
		

Links

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

Crossrefs

Cf. A001597, A006339, A016032, A025426, A266927, A273238.

Programs

Mathematica

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 *)

Extensions

a(9)-a(22) from Giovanni Resta, May 19 2016