This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A249573 #51 Mar 21 2017 10:41:57
%S A249573 2,3,13,19,373,174877,135859,18235423,26588257,93112729,376038903103,
%T A249573 7087694466289,120223669028389
%N A249573 Smallest prime p that remains prime through exactly n iterations of the function f(x) = 5x + 2.
%C A249573 Smallest p = prime(i) such that A280720(i) = n, where i is the index of p in A000040. - _Felix Fröhlich_, Jan 07 2017
%C A249573 From _Jon E. Schoenfield_, Jan 08 2017: (Start)
%C A249573 a(10) > 10^10.
%C A249573 It seems very likely that a(11) exists. But is it possible that this sequence is finite? Each row of the table below shows, for an interval of width 10^8, the number of primes p within the interval that remain prime through exactly 0 iterations, exactly 1 iteration, etc. E.g., in the interval [10^9, 10^9 + 10^8), there are 4437075 primes p that remain prime through exactly 0 iterations, 326699 that remain prime through exactly 1, 45062 that remain prime through exactly 2, etc.
%C A249573 ---------------------------------------------------------------
%C A249573 Fixed interval width = 10^8
%C A249573 ---------------------------------------------------------------
%C A249573 Start Number of successful iterations
%C A249573 of --------------------------------------------------------
%C A249573 intvl 0 1 2 3 4 5 6 7 8 9
%C A249573 ===== ======= ====== ===== ===== ==== ==== ==== ==== ==== ====
%C A249573 1 5225638 450798 73434 10139 1308 114 17 4 2 1
%C A249573 10^ 8 4858227 391247 59352 7720 841 84 9 1 1 0
%C A249573 10^ 9 4437075 326699 45062 5438 605 45 10 2 0 0
%C A249573 10^10 4031707 271882 34218 3722 367 30 3 1 0 0
%C A249573 10^11 3689861 228960 26414 2649 251 20 6 0 0 0
%C A249573 10^12 3400459 194999 20675 1973 158 17 1 0 0 0
%C A249573 10^13 3155004 168786 16699 1489 108 6 1 0 0 0
%C A249573 10^14 2940881 147025 13535 1153 81 4 0 0 0 0
%C A249573 10^15 2752985 128743 11275 874 55 5 0 0 0 0
%C A249573 ---------------------------------------------------------------
%C A249573 The numbers in column 0 drop at a rate that is not surprising, given that the way that the density of primes drops as numbers get larger. The numbers in the other columns drop more rapidly, in relative terms. Suppose a similar table were constructed using much wider intervals (perhaps with intervals starting not at 1, 10^8, 10^9, 10^10, etc., but at 1, 10^30, 10^31, 10^32, etc.), so that the numbers in, say, column 12 remained positive through several rows, but were dropping by a factor of more than 10 from one row to the next, making it likely that the total number of k-digit primes -- not just those from intervals of a fixed size -- that would remain prime through 12 iterations was actually decreasing as k increased. Would such an outcome suggest that the sequence might be finite? (End)
%e A249573 With p = 13: 5 * 13 + 2 = 67, 5 * 67 + 2 = 337 and 5 * 337 + 2 = 1687. 67 and 337 are both prime, but 1687 is not, so 13 remains prime through exactly two iterations of 5 * x + 2 and is the smallest prime with this property, so a(2) = 13.
%t A249573 c[p_] := Block[{k = 1, q = 5*p+2}, While[ PrimeQ[q], q = 5*q+2; k++]; k]; a[n_] := Block[{p = 2}, While[c[p] != n, p = NextPrime@ p]; p]; Array[a, 7] (* _Giovanni Resta_, Mar 21 2017 *)
%o A249573 (PARI) for(n=0, 10, forprime(p=2, 1e20, i=0; a=p; while(ispseudoprime(5*a+2), a=5*a+2; i++); if(i==n, print1(p, ", "); break(1))))
%Y A249573 Cf. A005602, A023217, A023313, A023341, A280720.
%K A249573 nonn,more
%O A249573 0,1
%A A249573 _Felix Fröhlich_, Nov 01 2014
%E A249573 a(10) from _Charles R Greathouse IV_, Jan 13 2017
%E A249573 a(11) from _John Cerkan_, Mar 20 2017
%E A249573 a(12) from _Giovanni Resta_, Mar 21 2017