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

5, 19, 31, 201829, 131681731, 954667531, 8998333416049

Offset: 1

Author

Abhiram R Devesh, Jun 02 2014

Keywords

• nonn
• hard
• more

Comments

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

A006512 : Primes p such that p - 2 is also prime.

A240126 : Primes p such that p - 2 and p^3 - 2 are also prime.

A242517 : Primes p such that p - 2, p^3 - 2 and p^5 - 2 are primes.

A242518 : Primes p such that p - 2, p^3 - 2, p^5 - 2 and p^7 - 2 are primes.

Examples

			For n = 1, p = 5, p - 2 = 3 is prime.
For n = 2, p = 19, p - 2 = 17 and p^3 - 2 = 6857 are primes.
For n = 3, p = 31, p - 2 = 29, p^3 - 2 = 29789, and p^5 - 2 = 28629149 are primes.
		

Crossrefs

Cf. A006512, A240126, A242517 and A242518.

Programs

Python

import sympy
## isp_list returns an array of true/false for prime number test for a
## list of numbers
def isp_list(ls):
    pt=[]
    for a in ls:
        if sympy.ntheory.isprime(a)==True:
            pt.append(True)
    return(pt)
co=1
while co < 7:
    al=0
    n=2
    while al!=co:
        d=[]
        for i in range(0, co):
            d.append(int(n**((2*i)+1))-2)
        al=isp_list(d).count(True)
        if al==co:
            ## Prints prime number and its corresponding sequence d
            print(n, d)
        n=sympy.ntheory.nextprime(n)
    co=co+1

Extensions

a(7) from Bert Dobbelaere, Aug 30 2020