A214089 Least prime p such that the first n primes divide p^2-1.

3, 5, 11, 29, 419, 1429, 1429, 315589, 1729001, 57762431, 1724478911, 6188402219, 349152569039, 1430083494841, 390499187164241, 1010518715554349, 18628320726623609, 522124211958421799, 522124211958421799, 5936798290039408015951, 311263131154464891496249

Offset: 1

Author

Robin Garcia, Jul 02 2012

Keywords

• nonn
• hard

Comments

(a(n)^2 - 1) / A002110(n) is congruent to 0 (mod 4). (a(n)^2 - 1) / (4*A002110(n)) = A215085(n). [J. Stauduhar, Aug 03 2012]

a(n) == +1 or -1 (mod prime(i)) for every i=1,2,...,n. The system of congruences x == +1 or -1 (mod prime(i)), i=1,2,...,n, has 2^(n-1) solutions modulo A002110(n) so that a(n) represents the smallest prime in the corresponding residue classes, allowing efficient computation (see PARI program). - Max Alekseyev, Aug 22 2012

Examples

			a(5) = 419: 419^2-1 = 175560 = 2^3*3*5*7*11*19 contains the first 5 primes.
a(7) = 1429:  1428=2^2*3*7*17, 1430=2*5*11*13 contains the first 7 primes.
a(8) = 315589: 315589^2-1 = 2^3*3*5*7*11*13*17^2*19*151 contains the first 8 primes.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..32

Crossrefs

Cf. A073917, A103783.

Programs

Maple

A214089 := proc(n)
     local m,k,p;
   m:= 2*mul(ithprime(j),j=1..n);
   for k from 1 do
     p:= sqrt(m*k+1);
     if type(p,integer) and isprime(p) then return(p)
     end if
   end do
end proc;
# Robert Israel, Aug 19 2012

Mathematica

f[n_] := Block[{k = 1, p = Times @@ Prime@Range@n}, While[! IntegerQ@Sqrt[4 k*p + 1], k++]; Block[{j = k}, While[! PrimeQ[Sqrt[4 j*p + 1]], j++]; Sqrt[4 j*p + 1]]]; Array[f, 10] (* J. Stauduhar, Aug 18 2012 *)

PARI

A214089(n) = {
        local(a,k=4,p) ;
        a=prod(j=1,n,prime(j)) ;
        while(1,
                if( issquare(k*a+1, &p),
                        if(isprime(p),
                                return(p);
                        ) ;
                ) ;
                k+=4;
        ) ;
} ;

PARI

{ a(n) = local(B,q); B=prod(i=1,n,prime(i))^2; forvec(v=vector(n-1,i,[0,1]), q=chinese(concat(vector(n-1,i,Mod((-1)^v[i],prime(i+1))),[Mod(1,2)])); forstep(s=lift(q),B-1,q.mod,if(ispseudoprime(s),B=s;break)) ); B } /* Max Alekseyev, Aug 22 2012 */

Python

from itertools import product
from sympy import sieve, prime, isprime
from sympy.ntheory.modular import crt
def A214089(n): return 3 if n == 1 else int(min(filter(isprime,(crt(tuple(sieve.primerange(prime(n)+1)), t)[0] for t in product((1,-1),repeat=n))))) # Chai Wah Wu, May 31 2022

Extensions

a(15)-a(16) from Donovan Johnson, Jul 25 2012

a(17) from Charles R Greathouse IV, Aug 08 2012

a(18) from Charles R Greathouse IV, Aug 16 2012

a(19) from J. Stauduhar, Aug 18 2012

a(20)-a(32) from Max Alekseyev, Aug 22 2012