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

1,1

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

Links

Table of n, a(n) for n=1..14.

Wikipedia, Wilson's theorem

Example

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.

Prog

(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)
}

Crossrefs

Cf. A051301, A230459

Sequence in context: A210109 A034546 A281584 * A228629 A010748 A272819

Adjacent sequences: A230312 A230313 A230314 * A230316 A230317 A230318

Keyword

nonn,hard,more

Author

James G. Merickel, Oct 15 2013

Extensions

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

Status

approved