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All terms except the first are congruent to 7, 11, 13 or 14 modulo 15. If we define
Pi(N,b) = # {p prime, p <= N, p == b (mod 15)};
Q(N) = # {p prime, 2 < p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ (94/95)*C*N/log(N) ~ (188/95)*C*(Pi(N,7) + Pi(N,11) + Pi(N,13) + Pi(N,14)), where C = A005596 is Artin's constant.
Conjecture: if we further define
Q(N,b) = # {p prime, p <= N, p == b (mod 15), p in this sequence},
then we have:
Q(N,7) ~ (10/47)*Q(N) ~ ( 80/95)*C*Pi(N,7);
Q(N,11) ~ (12/47)*Q(N) ~ ( 96/95)*C*Pi(N,11);
Q(N,13) ~ (10/47)*Q(N) ~ ( 80/95)*C*Pi(N,13);
Q(N,14) ~ (15/47)*Q(N) ~ (120/95)*C*Pi(N,14).
Numeric verification up tp N = 10^8:
| Q(N,7) | Q(N,11) | Q(N,13) | Q(N,14) | Q(N)
-------------+---------+---------+---------+---------+---------
N = 10^3 | 14 | 18 | 13 | 19 | 64
Q(N,*)/Q(N) | 0.21875 | 0.28125 | 0.20313 | 0.29688 | 1.00000
-------------+---------+---------+---------+---------+---------
N = 10^4 | 97 | 115 | 90 | 138 | 440
Q(N,*)/Q(N) | 0.22045 | 0.26136 | 0.20455 | 0.31364 | 1.00000
-------------+---------+---------+---------+---------+---------
N = 10^5 | 753 | 891 | 750 | 1129 | 3523
Q(N,*)/Q(N) | 0.21374 | 0.25291 | 0.21289 | 0.32047 | 1.00000
-------------+---------+---------+---------+---------+---------
N = 10^6 | 6153 | 7395 | 6176 | 9247 | 28971
Q(N,*)/Q(N) | 0.21238 | 0.25526 | 0.21318 | 0.31918 | 1.00000
-------------+---------+---------+---------+---------+---------
N = 10^7 | 52427 | 62973 | 52368 | 78398 | 246166
Q(N,*)/Q(N) | 0.21297 | 0.25582 | 0.21273 | 0.31848 | 1.00000
-------------+---------+---------+---------+---------+---------
N = 10^8 | 453936 | 544551 | 453699 | 680226 | 2132412
Q(N,*)/Q(N) | 0.21287 | 0.25537 | 0.21276 | 0.31899 | 1.00000
-------------+---------+---------+---------+---------+---------
Conjectured | 0.21277 | 0.25532 | 0.21277 | 0.31915 | 1.00000
(End)
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