A243817 -id:A243817 - cp's OEIS frontend

A243818 Primes p for which p^i - 4 is prime for i = 1, 3 and 5.

11, 971, 1877, 2861, 8741, 12641, 13163, 16763, 28283, 29021, 30707, 36713, 41957, 42227, 58967, 98717, 105971, 115127, 128663, 138641, 160817, 164093, 167441, 190763, 205607, 210173, 211067, 228341, 234197, 237977, 246473, 249107, 276557, 295433, 312233

Offset: 1

Abhiram R Devesh, Jun 11 2014

This is a subsequence of the following:
A046132: Larger member p+4 of cousin primes (p, p+4).
A243817: Primes p for which p - 4 and p^3 - 4 are primes.

			p = 11 is in this sequence because p - 4 = 7  (prime), p^3 - 4 = 1327 (prime) and p^5 - 4 = 161047 (prime).
p = 971 is in this sequence because p - 4 = 967  (prime), p^3 - 4 = 915498607 (prime) and p^5 - 4 = 863169625893847 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..250 from Abhiram R Devesh)

Cf. A046132, A243817.

Select[Range[300000], PrimeQ[#] && AllTrue[#^{1, 3, 5} - 4, PrimeQ] &] (* Amiram Eldar, Apr 04 2020 *)
Select[Prime[Range[27000]],AllTrue[#^{1,3,5}-4,PrimeQ]&] (* Harvey P. Dale, Jan 04 2021 *)

import sympy.ntheory as snt
n=5
while n>1:
    n1=n-4
    n2=((n**3)-4)
    n3=((n**5)-4)
    ##Check if n1 , n2 and n3 are also primes.
    if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True:
        print(n, n1, n2, n3)
    n=snt.nextprime(n)

A243885 Smallest prime p_n which generates n primes of the form (p_n^i - 4) when i runs through the first n odd numbers.

Original entry on oeis.org

7, 11, 11, 971, 71394923, 959316767, 13342820302307

Offset: 1

Abhiram R Devesh, Jun 13 2014

The first 4 entries of this sequence are the first entry of the following sequences:
A046132 : Larger member p+4 of cousin primes (p, p+4).
A243817 : Primes p for which p - 4 and p^3 - 4 are primes.
A243818 : Primes p for which p^i - 4 is prime for i = 1, 3 and 5.
A243861 : Primes p for which p^i - 4 is prime for i = 1, 3, 5 and 7.

			a(1) = 7 because 7-4 = 3 (prime),
a(2) = 11 because 11-4 = 7 (prime) and  11^3 - 4 = 1327 (prime).

Cf. A046132, A243817, A243818 and A243861.

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 > 0:
    al=0
    n=2
    while al!=co:
        d=[]
        for i in range(0, co):
            d.append(int(n**((2*i)+1))-4)
        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

a(6) from Bert Dobbelaere, Jul 16 2019

a(7) from Giovanni Resta, Jul 18 2019

cp's OEIS Frontend

A243818 Primes p for which p^i - 4 is prime for i = 1, 3 and 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python