There appear to be at least n primes in the range (x-2*sqrt(x), x] for all x >= a(n).

Terms

a(0) =2a(1) =3a(2) =37a(3) =139a(4) =331a(5) =1409a(6) =1423a(7) =1427a(8) =2239a(9) =3163a(10) =3181a(11) =3511a(12) =6547a(13) =7433a(14) =7457a(15) =7487a(16) =10061a(17) =11777a(18) =11779a(19) =14401a(20) =18899a(21) =19081a(22) =19373a(23) =23537a(24) =24763a(25) =27617a(26) =27673a(27) =32027a(28) =32051a(29) =38113