A230315 a(n) is the smallest prime dividing n numbers of the form k! + 1.

2, 7, 23, 59, 71, 71, 71, 3643, 62939, 292627, 292627, 1089427, 2374649, 2374649

Offset: 1

Author

James G. Merickel, Oct 15 2013

Keywords

• nonn
• hard
• more

Comments

Note that were the title altered to instead count the numbers k for which p divides k!+1, then a(2) would be 2 rather than 7 (as 0!+1=1!+1).

Ties arise for the following list of values: 3, 5, 11, 19, 61, 661, 2267, 3163, 3541, 6529, 9697, 12227, 40751, 46687, 51347, 59447, 69493, 72077, 72923, 83579, 141907, 167267, 201667 and 212207 (and were not sought beyond a(11)).

Search for a(15) completed through the 260000th prime. - James G. Merickel, Jan 16 2014

a(15) > 1.1*10^8. - Giovanni Resta, Jun 10 2018

Examples

			71 divides 7!+1, 9!+1, 19!+1, 51!+1, 61!+1, 63!+1, and of course 70!+1 (Wilson's Theorem).  Since a(4)=59 and 61 and 67 do not enter in, 71=a(n) for n=5 to 7.
		

Links

• Wikipedia, Wilson's theorem

Crossrefs

Cf. A051301, A230459

Programs

PARI

{
\\ y is an arbitrary value. \\
rec=0;y=10^7;z=primepi(y);a=vector(z,x,1);
b=vector(z);q=vector(z,x,prime(x));i=1;
for(k=1,z,
  for(r=i,q[k],
    for(j=k,z,
      a[j]*=r;a[j]%=q[j];
      if(a[j]==q[j]-1,b[j]++));
    while(b[j]>rec,
      rec++;print1(q[j]", ")));
  i=q[k]+1)
}

Extensions

a(12)-a(14) added (with a search limit for a(15) in Comments) by James G. Merickel, Jan 16 2014