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A328686 Define a map from the primes to the primes by f(p) = (p-1)/2 if that is prime, or else (p+1)/2 if that is prime, and otherwise is undefined. Start with the n-th prime and iterate f until we cannot go any further; a(n) is the number of steps.

%I A328686 #34 Dec 10 2019 17:45:27
%S A328686 0,1,1,2,2,3,0,0,3,0,0,1,0,0,4,0,1,1,0,0,2,0,1,0,0,0,0,1,0,0,0,0,0,0,
%T A328686 0,0,1,0,2,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,2,0,0,0,
%U A328686 1,0,0,2,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1
%N A328686 Define a map from the primes to the primes by f(p) = (p-1)/2 if that is prime, or else (p+1)/2 if that is prime, and otherwise is undefined. Start with the n-th prime and iterate f until we cannot go any further; a(n) is the number of steps.
%C A328686 For each prime, the end of the trajectory is reached when one cannot generate another prime number from it.
%C A328686 For example, p(3) = 5 -> 2 (1 iteration), so a(3)=1. Also p(5) = 11 -> 5 -> 2 (2 iterations), 23 -> 11 -> 5 -> 2 (3 iterations) and 47 -> 23 -> 11 -> 5 -> 2 (4 iterations). Hence a(3) = 1, a(5) = 2, a(9) = 3 and a(15) = 4.
%C A328686 a(n) = 0 for n = 1, 7, 8, 10, 11, 13, 14, 16, 19, 20, 22, 24, 25, ... The corresponding primes are A176902(n) = 2, 17, 19, 29, 31, 41, 43, ... .
%C A328686 The sequence of the last terms of the trajectories begins with 2, 2, 2, 2, 2, 2, 17, 19, 2, 29, 31, 19, 41, 43, 2, 53, 29, 31, 67, ...
%C A328686 The following table gives the trajectories of the smallest prime requiring 0, 1, 2, 3, 4, 5, 6, iterations:
%C A328686 +------------+----------+------------------------------------------+
%C A328686 | Number of  | smallest |               trajectory                 |
%C A328686 | iterations |  prime   |                                          |
%C A328686 +------------+----------+------------------------------------------+
%C A328686 |      0     |       2  |  2                                       |
%C A328686 |      1     |       3  |  3 -> 2                                  |
%C A328686 |      2     |       7  |  7 -> 3 -> 2                             |
%C A328686 |      3     |      13  | 13 -> 7 -> 3 -> 2                        |
%C A328686 |      4     |      47  | 47 -> 23 -> 11 -> 5 -> 2                 |
%C A328686 |      5     |    2879  | 2879 -> 1439 -> 719 -> 359 -> 179 -> 89  |
%C A328686 |      6     | 1065601  | 1065601 -> 532801 -> 266401 -> 133201 -> |
%C A328686 |            |          |   66601 -> 33301 -> 16651                |
%C A328686 +------------+----------+------------------------------------------+
%e A328686 a(15) = 4 because prime(15) = 47 and 47 -> 23 -> 11 -> 5 -> 2 with 4 iterations.
%p A328686 for n from 1 to 100 do:
%p A328686    ii:=0:it:=0:p:=ithprime(n):
%p A328686    for i from 1 to 100 while(ii=0)  :
%p A328686      p1:=(p-1)/2:p2:=(p+1)/2:
%p A328686       if type(p1,prime)=false and type(p2,prime)=false
%p A328686        then
%p A328686        ii:=1:printf(`%d, `,it):
%p A328686        else
%p A328686        it:=it+1:
%p A328686         if isprime(p1)
%p A328686          then
%p A328686           p:=p1:
%p A328686           else
%p A328686           p:=p2:
%p A328686          fi:
%p A328686          fi:
%p A328686         od:
%p A328686        od:
%t A328686 f[p_] := If[PrimeQ[(q = (p-1)/2)], q, If[PrimeQ[(r = (p+1)/2)], r, 0]]; g[n_] := -2 + Length @ NestWhileList[f, n, #>0 &]; g /@ Select[Range[457], PrimeQ] (* _Amiram Eldar_, Nov 16 2019 *)
%Y A328686 Cf. A000040, A005383, A005385, A176902.
%Y A328686 The underlying map is A330310.
%K A328686 nonn
%O A328686 1,4
%A A328686 _Michel Lagneau_, Oct 25 2019