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Primes which are the sums of two consecutive nonprimes ( A141468).
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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
COMMENTS
Five together with primes that are the sum of two consecutive composite numbers.
MATHEMATICA
2*Select[Range[300], !PrimeQ[#] == !PrimeQ[#+1] && PrimeQ[2*#+1] &] + 1 (* G. C. Greubel, Jul 01 2016; Nov 10 2023 *)
PROG
(Magma) [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 (SageMath) [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
EXTENSIONS
Typo corrected and terms checked by D. S. McNeil, Nov 17 2010
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.
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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
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 *)
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