A247965 a(n) is the smallest number k such that m*k^2+1 is prime for all m = 1 to n.

1, 1, 6, 3240, 113730, 30473520, 3776600100, 16341921960, 3332396388090

Offset: 1

Author

Michel Lagneau, Sep 28 2014

Keywords

• nonn
• hard
• more

Comments

Conjecture : the sequence is infinite.

a(10) > 15466500000000. a(11) > 107669100000000. - Hiroaki Yamanouchi, Oct 01 2014

Examples

			a(3)=6 because 6^2+1 = 37, 2*6^2+1 = 73 and 3*6^2+1 = 109 are prime numbers.
The resulting primes begin like this:
2;
2, 3;
37, 73, 109;
10497601, 20995201, 31492801, 41990401;
... - _Michel Marcus_, Sep 29 2014
		

Crossrefs

Cf. A002496, A090698, A002648, A137530, A090687, A201602, A090685, A156226.

Programs

Maple

for n from 1 to 6 do:
  ii:=0:
   for k from 1 to 10^10 while(ii=0) do:
     ind:=0:
       for m from 1 to n do:
         p:=m*k^2+1:
          if type(p,prime) then
           ind:=ind+1:
           fi:
        od:
       if ind=n then
        ii:=1:printf ( "%d %d \n",n,k):
       fi:
    od:
  od:

PARI

a(n)=k=1;while(k,c=0;for(i=1,n,if(!ispseudoprime(i*k^2+1),c++;break));if(!c,return(k));if(c,k++))
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Sep 28 2014

Extensions

a(7)-a(9) from Hiroaki Yamanouchi, Oct 01 2014