A161665 Primes that can be represented as a sum of 2 and also as a sum of 3 distinct nonzero squares, sharing a term in the sums.

29, 101, 109, 149, 173, 181, 229, 233, 241, 269, 293, 389, 401, 409, 421, 433, 449, 521, 569, 641, 661, 677, 701, 757, 761, 769, 797, 821, 857, 877, 881, 941, 1021, 1069, 1097, 1109, 1117, 1181, 1229, 1237, 1277, 1289, 1301, 1373, 1381, 1429, 1433, 1481, 1549

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jun 15 2009

Keywords

• nonn

Comments

Dropping the requirement of one shared term, we would get the supersequence 17, 29, 41, 53, 61, 73, ... - R. J. Mathar, Oct 04 2009

Examples

			The prime 29 has the representations 29 = 2^2+ 5^2 = 2^2+3^2+4^2, sharing 2^2.
The prime 101 has the representations 101 = 1^2+10^2 = 1^2+6^2+8^2, sharing 1^2.
The prime 109 has the representations 109 = 3^2+10^2 = 3^2+6^2+8^2, sharing 3^2.
The prime 149 has the representations 149 = 7^2+10^2 = 6^2+7^2+8^2, sharing 7^2.
		

Crossrefs

Cf. A002114, A085317.

Programs

Mathematica

f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[a=f[n];If[a>0,b=f[n-(f[n])^2]; If[b>0,c=(n-a^2-b^2)^(1/2);If[a!=b&&a!=c,If[PrimeQ[n],AppendTo[lst, n]]]]],{n,3,4*6!}];lst

Extensions

Definition reverse-engineered from program by R. J. Mathar, Oct 04 2009