A242886 Smallest prime p_n which generates n primes of the form (p^i + 2) where i represents the first n odd numbers.

3, 3, 419, 132749, 514664471, 1164166301, 364231372931

Offset

1,1

Comments

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

a. A001359: Lesser of twin primes.

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

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

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

a(8) > 10^14. - Bert Dobbelaere, Aug 31 2020

Links

Table of n, a(n) for n=1..7.

Example

For n = 1, p = 3 generates primes of the form p^n + 2; for i = 1,
   p + 2 = 5 (prime).
For n = 2, p = 3 generates primes of the form p^n + 2; for i = 1 and 3,
   p + 2 = 5 (prime) and p^3 + 2 = 29 (prime).
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).

Prog

(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

Crossrefs

Cf. A001359, A240110, A242326, A242327.

Sequence in context: A344283 A335258 A292163 * A221947 A138662 A009011

Adjacent sequences: A242883 A242884 A242885 * A242887 A242888 A242889

Keyword

nonn,hard,more

Author

Abhiram R Devesh, May 25 2014

Extensions

a(7) from Bert Dobbelaere, Aug 30 2020

Status

approved