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a(1)=3,a(2)=5; a(n+1)=smallest prime number > a(n-1) such that the sum of any three consecutive terms is a prime.
1

%I #2 Mar 31 2012 12:38:27

%S 3,5,5,7,7,17,13,23,17,31,19,47,23,61,29,67,31,83,37,103,41,107,43,

%T 113,67,127,83,137,97,139,101,149,103,157,107,167,109,173,127,179,137,

%U 193,149,199,151,227,163,229,179,233,181,239,193,241,197,263,199,271,239

%N a(1)=3,a(2)=5; a(n+1)=smallest prime number > a(n-1) such that the sum of any three consecutive terms is a prime.

%t a=3;b=5;lst={a,b};Do[Do[If[PrimeQ[q]&&PrimeQ[a+b+q],c=q;Break[]],{q,a+2,9!,2}];AppendTo[lst,c];a=b;b=c,{n,6!}];lst

%Y Cf. A062391

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Nov 22 2009