A214149 Least prime p such that the factorization of p^2-9 contains n consecutive primes beginning with prime(3)=5.

7, 17, 157, 283, 20023, 20023, 6446437, 14382547, 122862737, 12925003913, 625586209427, 761375971073, 92757861866387, 15447055149567577, 192604162645538927, 192604162645538927, 724012906264106939197, 2667069644892918607163, 235168333030918497994787

Offset: 1

Author

Robin Garcia, Jul 05 2012

Keywords

• nonn

Comments

We consider prime-smoothness for primes >=5, because primes p>3 are not divisible by 3, and so p-3 and p+3 are not divisible by 3.

Examples

			20020 = 2^2*5*7*11*13, 20026 = 2*17*19*31; 20023^2-9 contains 6 all-consecutive primes beginning with 5.
6446437^2-9 = 2^4*5*7^2*11*13*17^2*19*23*587 contains 7 all-consecutive primes, the first one being 5.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..29

Crossrefs

Cf. A214089, A214150.

Programs

PARI

A214149(n)={ local(a, k=1, p) ; a=prod(j=3, n+2, prime(j)) ; while(1, if( issquare(k*a+9), p=sqrtint(k*a+9) ; if(isprime(p),return(p); ) ;  ) ; k++ ; ) }

Python

from itertools import product
from sympy import isprime, sieve, prime
from sympy.ntheory.modular import crt
def A214149(n): return 7 if n == 1 else int(min(filter(lambda n: n > 3 and isprime(n),(crt(tuple(sieve.primerange(5,prime(n+2)+1)), t)[0] for t in product((3,-3),repeat=n))))) # Chai Wah Wu, Jun 01 2022

Extensions

More terms from Max Alekseyev, Aug 22 2012