login
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.
1
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 |
+------------+----------+------------------------------------------+
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
The underlying map is A330310.
Sequence in context: A338700 A246471 A289814 * A330622 A330629 A304760
KEYWORD
nonn
AUTHOR
Michel Lagneau, Oct 25 2019
STATUS
approved