A226640 a(n) = 2^x+2^y where p(n) is the n-th prime of the form 4*k+1 and x, y is the unique integer solution to p(n) = x^2+y^2.

6, 12, 18, 36, 66, 48, 132, 96, 264, 288, 528, 1026, 1032, 384, 2064, 1152, 2112, 8196, 1536, 4224, 16386, 32772, 8448, 32784, 65538, 9216, 16896, 65568, 131076, 12288, 18432, 66048, 262176, 131328, 262272, 132096, 524352, 1048578, 1048584

Offset: 1

Author

Frank M Jackson, Aug 19 2013

Keywords

• nonn

Comments

Gauss proved that any prime of the form 4*k+1 (A002144) is equal to the unique sum of two squares. This sequence identifies these summed squares and uniquely maps them into a decimal from which the two squares can be retrieved. The mapping is given by a(n) = 2^x+2^y where p(n) = x^2+y^2.

Examples

			p(6) = 41 and 41 = 4^2 + 5^2 hence a(6) = 2^4+2^5 = 48. To retrieve the values 4 and 5 from 48 convert 48 to binary. The 1 bits (there are only ever two) select 2^4 and 2^5. So x, y are 4, 5.
		

Links

• L. A. Butler, A Classification of Gaussian Primes, School of Mathematics, Bristol Univ.

Crossrefs

Cf. A002144

Programs

Mathematica

next1m4prime[n1_] := (n2=n1+1; While[!PrimeQ[n2]||!Mod[n2, 4]==1, n2++]; n2); getbinmap[m1_] := (m2=m1; m3=Floor[Sqrt[m2]]; While[!IntegerQ@Sqrt[m2-m3^2], m3--]; 2^Sqrt[m2-m3^2] + 2^m3); SetAttributes[getbinmap, Listable]; getbinmap[Table[Nest[next1m4prime, 1, n], {n, 1, 100}]]