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%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
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