Let b(0)=1/2, b(n) = (b(n-1)+Prime[n])/2; sequence gives 2^(n+1)*b(n).

Terms

a(0) =1a(1) =5a(2) =17a(3) =57a(4) =169a(5) =521a(6) =1353a(7) =3529a(8) =8393a(9) =20169a(10) =49865a(11) =113353a(12) =264905a(13) =600777a(14) =1305289a(15) =2845385a(16) =6318793a(17) =14052041a(18) =30042825a(19) =65170121a(20) =139619017a(21) =292711113a(22) =624061129a(23) =1320315593a(24) =2813487817a(25) =6068267721a(26) =12846262985a(27) =26670688969