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

1,4

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 |

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

Links

Table of n, a(n) for n=1..88.

Example

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

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 *)

Crossrefs

Cf. A000040, A005383, A005385, A176902.

The underlying map is A330310.

Sequence in context: A338700 A246471 A289814 * A330622 A330629 A304760

Adjacent sequences: A328683 A328684 A328685 * A328687 A328688 A328689

Keyword

nonn

Author

Michel Lagneau, Oct 25 2019

Status

approved