1 + Sum_{k=2..n} 2^((prime(k)-1)/2).

Terms

a(0) =1a(1) =3a(2) =7a(3) =15a(4) =47a(5) =111a(6) =367a(7) =879a(8) =2927a(9) =19311a(10) =52079a(11) =314223a(12) =1362799a(13) =3459951a(14) =11848559a(15) =78957423a(16) =615828335a(17) =1689570159a(18) =10279504751a(19) =44639243119a(20) =113358719855a(21) =663114533743