%I
%S 5,11,17,37,53,89,107,127,179,197,223,233,257,263,401,409,421,449,457,
%T 661
%N a(n) is the least prime p such that p is greater than any previous term and is representable as the arithmetic mean of two other primes in exactly n different ways.
%C A number p is representable as the arithmetic mean of two other primes in n ways if there are n values of k such that p + k and p  k are both prime.
%C If the restriction that a(n) must be greater than previous terms is removed then the sequence would be A126204.  _Andrew Howroyd_, Jan 12 2020
%C The next term if it exists > 10^6.  _Andrew Howroyd_, Jan 12 2020
%e a(1) = 5 because 5+2 are primes.
%e a(2) = 11 because 11+6, 11+8 are primes.
%e a(3) = 17 because 17+6, 17+12, 17+=14 are primes.
%e a(4) = 37 because 37+6, 37+24, 37+30, 37+34 are primes.
%e a(5) = 53 because 53+6, 53+30, 53+36, 53+48, 53+50 are primes.
%t q=1;lst={};Do[p=Prime[n];i=0;Do[If[PrimeQ[pk]&&PrimeQ[p+k],i++;],{k,2,p,2}];If[i==q,AppendTo[lst,p];q++ ],{n,2*5!}];lst
%o (PARI) a(n, lim=oo)={my(v=vector(n),r=1); forprime(p=5, lim, my(k=0); forprime(q=3, p2, k+=isprime(2*pq)); if(k==r, if(r==n, return(p)); r++))} \\ _Andrew Howroyd_, Jan 12 2020
%Y Cf. A126204.
%K nonn,more
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Mar 01 2009
%E Definition clarified by _Andrew Howroyd_, Jan 12 2020
