p, p+6 and p+8 are all primes (A046138) but p+2 is not.

Terms

a(0) =23a(1) =53a(2) =131a(3) =173a(4) =233a(5) =263a(6) =563a(7) =593a(8) =653a(9) =1013a(10) =1223a(11) =1283a(12) =1601a(13) =1613a(14) =2333a(15) =2543a(16) =2963a(17) =3323a(18) =3533a(19) =3761a(20) =3911a(21) =3923a(22) =4013a(23) =4211a(24) =4253a(25) =4643a(26) =4793a(27) =5003a(28) =5273a(29) =5471