A176902 -id:A176902 - cp's OEIS frontend

A167915 Primes which are the sums of two consecutive nonprimes (A141468).

5, 17, 19, 29, 31, 41, 43, 53, 67, 71, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 163, 173, 181, 191, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 281, 283, 293, 307, 311, 317, 331, 337, 349, 353, 367, 373, 379, 389, 401, 409

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2009

nonn

Five together with primes that are the sum of two consecutive composite numbers.

			a(1)=1+4=5, a(2)=8+9=17.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A060254, A141468, A176902.

[2*n+1: n in [1..300] | (not IsPrime(n) eq not IsPrime(n+1)) and IsPrime(2*n+1)]; // G. C. Greubel, Nov 10 2023

2*Select[Range[300], !PrimeQ[#] == !PrimeQ[#+1] && PrimeQ[2*#+1] &] + 1 (* G. C. Greubel, Jul 01 2016; Nov 10 2023 *)

[2*n+1 for n in (1..300) if  (not is_prime(n)) - (not is_prime(n+1)) == 0 and is_prime(2*n+1)] # G. C. Greubel, Nov 10 2023

a(n+1) = A060254(n) = A176902(n+1). - Juri-Stepan Gerasimov, Apr 28 2010

Typo corrected and terms checked by D. S. McNeil, Nov 17 2010

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.

Original entry on oeis.org

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

Michel Lagneau, Oct 25 2019

nonn

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                |
+------------+----------+------------------------------------------+

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

Cf. A000040, A005383, A005385, A176902.

The underlying map is A330310.

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:

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 *)

cp's OEIS Frontend

A167915 Primes which are the sums of two consecutive nonprimes (A141468).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions