A051301 -id:A051301 - cp's OEIS frontend

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

James G. Merickel, Oct 15 2013

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

			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.

Wikipedia, Wilson's theorem

Cf. A051301, A230459

{
\\ 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)
}

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

A230459 Ordered by increasing m with k < m, a(n) is the n-th record value of gcd(k!+1, m!+1).

Original entry on oeis.org

2, 7, 71, 661, 733, 2371, 3529, 13499, 46549, 98101, 163517, 197933, 1924217, 3322441, 5370731

Offset: 1

James G. Merickel, Oct 19 2013

The pairs (m,k) generating records are (1,0), (6,3), (9,7), (17,8), (89,51), (174,144), (349,228), (422,81), (650,406), (1415,1718), (1697,161), (1622,773), (1884,1219), (7003,2031) and (17057,660).
Heuristics in concert with a database of 'small' (less than, say, 10^12) prime factors of numbers of this kind would generate faster accurate results with near certainty, while any truly proving program is doomed to be relatively slow by comparison (and see following on a(15)).
Note: An auxiliary program employed a limit of 10^8--in lieu of a database--to generate the likely-but-not-certain value of a(15) shown last.

			a(1)=2, corresponding to m=1 and k=0.  7 is the first value other than 1 to be the greatest common divisor of two different numbers k!+1 and m!+1, where m is increasing and k is allowed to increase to m-1 for a given m (for m=6 and k=3, m!+1=7*103 and k!+1=7); so that a(2)=7.

Cf. A002583, A002981, A038507, A051301.

{
\\ The constant L here is arbitrary.\\
\\ This does not generate a(1).\\
rec=2;L=10000;F=vector(L);n=2;
for(k=1,L,n--;n*=k;n++;F[k]=n);
for(m=2,L,
  for(k=1,m-1,
    a=gcd(F[m],F[k]);if(a>rec,
      rec=a;print1(a": "m","k"\n"))))
}

cp's OEIS Frontend

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

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