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.

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, 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, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Michel Lagneau, Oct 25 2019

Keywords

• nonn

Comments

For each prime, the end of the trajectory is reached when one cannot generate another prime number from it.

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.

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, ... .

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, ...

The following table gives the trajectories of the smallest prime requiring 0, 1, 2, 3, 4, 5, 6, iterations:

+------------+----------+------------------------------------------+

| Number of | smallest | trajectory |

| iterations | prime | |

+------------+----------+------------------------------------------+

| 0 | 2 | 2 |

| 1 | 3 | 3 -> 2 |

| 2 | 7 | 7 -> 3 -> 2 |

| 3 | 13 | 13 -> 7 -> 3 -> 2 |

| 4 | 47 | 47 -> 23 -> 11 -> 5 -> 2 |

| 5 | 2879 | 2879 -> 1439 -> 719 -> 359 -> 179 -> 89 |

| 6 | 1065601 | 1065601 -> 532801 -> 266401 -> 133201 -> |

| | | 66601 -> 33301 -> 16651 |

+------------+----------+------------------------------------------+

Examples

			a(15) = 4 because prime(15) = 47 and 47 -> 23 -> 11 -> 5 -> 2 with 4 iterations.
		

Crossrefs

Cf. A000040, A005383, A005385, A176902.

The underlying map is A330310.

Programs

Maple

for n from 1 to 100 do:
   ii:=0:it:=0:p:=ithprime(n):
   for i from 1 to 100 while(ii=0)  :
     p1:=(p-1)/2:p2:=(p+1)/2:
      if type(p1,prime)=false and type(p2,prime)=false
       then
       ii:=1:printf(`%d, `,it):
       else
       it:=it+1:
        if isprime(p1)
         then
          p:=p1:
          else
          p:=p2:
         fi:
         fi:
        od:
       od:

Mathematica

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 *)