A249573 Smallest prime p that remains prime through exactly n iterations of the function f(x) = 5x + 2.

2, 3, 13, 19, 373, 174877, 135859, 18235423, 26588257, 93112729, 376038903103, 7087694466289, 120223669028389

Offset: 0

Author

Felix Fröhlich, Nov 01 2014

Keywords

• nonn
• more

Comments

Smallest p = prime(i) such that A280720(i) = n, where i is the index of p in A000040. - Felix Fröhlich, Jan 07 2017

From Jon E. Schoenfield, Jan 08 2017: (Start)

a(10) > 10^10.

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.

---------------------------------------------------------------

Fixed interval width = 10^8

---------------------------------------------------------------

Start Number of successful iterations

of --------------------------------------------------------

intvl 0 1 2 3 4 5 6 7 8 9

===== ======= ====== ===== ===== ==== ==== ==== ==== ==== ====

1 5225638 450798 73434 10139 1308 114 17 4 2 1

10^ 8 4858227 391247 59352 7720 841 84 9 1 1 0

10^ 9 4437075 326699 45062 5438 605 45 10 2 0 0

10^10 4031707 271882 34218 3722 367 30 3 1 0 0

10^11 3689861 228960 26414 2649 251 20 6 0 0 0

10^12 3400459 194999 20675 1973 158 17 1 0 0 0

10^13 3155004 168786 16699 1489 108 6 1 0 0 0

10^14 2940881 147025 13535 1153 81 4 0 0 0 0

10^15 2752985 128743 11275 874 55 5 0 0 0 0

---------------------------------------------------------------

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)

Examples

			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.
		

Crossrefs

Cf. A005602, A023217, A023313, A023341, A280720.

Programs

Mathematica

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 *)

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))))

Extensions

a(10) from Charles R Greathouse IV, Jan 13 2017

a(11) from John Cerkan, Mar 20 2017

a(12) from Giovanni Resta, Mar 21 2017