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A243269 Smallest prime p such that p^k - 2 is prime for all odd exponents k from 1 up to 2*n-1 (inclusive).

%I A243269 #22 Sep 24 2024 14:59:32
%S A243269 5,19,31,201829,131681731,954667531,8998333416049
%N A243269 Smallest prime p such that p^k - 2 is prime for all odd exponents k from 1 up to 2*n-1 (inclusive).
%C A243269 The first 4 entries of this sequence are the first entry of the following sequences:
%C A243269 A006512 : Primes p such that p - 2 is also prime.
%C A243269 A240126 : Primes p such that p - 2 and p^3 - 2 are also prime.
%C A243269 A242517 : Primes p such that p - 2, p^3 - 2 and p^5 - 2 are primes.
%C A243269 A242518 : Primes p such that p - 2, p^3 - 2, p^5 - 2 and p^7 - 2 are primes.
%e A243269 For n = 1, p = 5, p - 2 = 3 is prime.
%e A243269 For n = 2, p = 19, p - 2 = 17 and p^3 - 2 = 6857 are primes.
%e A243269 For n = 3, p = 31, p - 2 = 29, p^3 - 2 = 29789, and p^5 - 2 = 28629149 are primes.
%o A243269 (Python)
%o A243269 import sympy
%o A243269 ## isp_list returns an array of true/false for prime number test for a
%o A243269 ## list of numbers
%o A243269 def isp_list(ls):
%o A243269     pt=[]
%o A243269     for a in ls:
%o A243269         if sympy.ntheory.isprime(a)==True:
%o A243269             pt.append(True)
%o A243269     return(pt)
%o A243269 co=1
%o A243269 while co < 7:
%o A243269     al=0
%o A243269     n=2
%o A243269     while al!=co:
%o A243269         d=[]
%o A243269         for i in range(0, co):
%o A243269             d.append(int(n**((2*i)+1))-2)
%o A243269         al=isp_list(d).count(True)
%o A243269         if al==co:
%o A243269             ## Prints prime number and its corresponding sequence d
%o A243269             print(n, d)
%o A243269         n=sympy.ntheory.nextprime(n)
%o A243269     co=co+1
%Y A243269 Cf. A006512, A240126, A242517 and A242518.
%K A243269 nonn,hard,more
%O A243269 1,1
%A A243269 _Abhiram R Devesh_, Jun 02 2014
%E A243269 a(7) from _Bert Dobbelaere_, Aug 30 2020