A242886 - OEIS

%I #24 Aug 31 2020 02:42:16

%S 3,3,419,132749,514664471,1164166301,364231372931

%N Smallest prime p_n which generates n primes of the form (p^i + 2) where i represents the first n odd numbers.

%C The first 4 entries of this sequence are the first entry of the following sequences:

%C a. A001359: Lesser of twin primes.

%C b. A240110: Primes p such that p + 2 and p^3 + 2 are also prime.

%C c. A242326: Primes p for which p + 2, p^3 + 2, and p^5 + 2 are also prime.

%C d. A242327: Primes p for which (p^n) + 2 is prime for n = 1, 3, 5, and 7.

%C a(8) > 10^14. - _Bert Dobbelaere_, Aug 31 2020

%e For n = 1, p = 3 generates primes of the form p^n + 2; for i = 1,

%e    p + 2 = 5 (prime).

%e For n = 2, p = 3 generates primes of the form p^n + 2; for i = 1 and 3,

%e    p + 2 = 5 (prime) and p^3 + 2 = 29 (prime).

%e For n = 3, p = 419 generates primes of the form p^n + 2; for i = 1, 3, and  5, p + 2 = 421 (prime), p^3 + 2 = 73560061 (prime), and p^5 + 2 = 12914277518101 (prime).

%o (Python)

%o import sympy

%o ## isp_list returns an array of true/false for prime number test for a

%o ## list of numbers

%o def isp_list(ls):

%o ....pt=[]

%o ....for a in ls:

%o ........if sympy.ntheory.isprime(a)==True:

%o ............pt.append(True)

%o ....return(pt)

%o co=1

%o while co < 7:

%o ....al=0

%o ....n=2

%o ....while al!=co:

%o ........d=[]

%o ........for i in range(0,co):

%o ............d.append(int(n**((2*i)+1))+2)

%o ........al=isp_list(d).count(True)

%o ........if al==co:

%o ............## Prints prime number and its corresponding sequence d

%o ............print(n,d)

%o ........n=sympy.ntheory.nextprime(n)

%o ....co=co+1

%Y Cf. A001359, A240110, A242326, A242327.

%K nonn,hard,more

%O 1,1

%A _Abhiram R Devesh_, May 25 2014

%E a(7) from _Bert Dobbelaere_, Aug 30 2020