A230315 - OEIS

%I #34 Jun 10 2018 16:05:24

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

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

%C 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).

%C 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)).

%C Search for a(15) completed through the 260000th prime. - _James G. Merickel_, Jan 16 2014

%C a(15) > 1.1*10^8. - _Giovanni Resta_, Jun 10 2018

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Wilson%27s_theorem">Wilson's theorem</a>

%e 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.

%o (PARI)

%o {

%o \\ y is an arbitrary value. \\

%o rec=0;y=10^7;z=primepi(y);a=vector(z,x,1);

%o b=vector(z);q=vector(z,x,prime(x));i=1;

%o for(k=1,z,

%o   for(r=i,q[k],

%o     for(j=k,z,

%o       a[j]*=r;a[j]%=q[j];

%o       if(a[j]==q[j]-1,b[j]++));

%o     while(b[j]>rec,

%o       rec++;print1(q[j]", ")));

%o   i=q[k]+1)

%o }

%Y Cf. A051301, A230459

%K nonn,hard,more

%O 1,1

%A _James G. Merickel_, Oct 15 2013

%E a(12)-a(14) added (with a search limit for a(15) in Comments) by _James G. Merickel_, Jan 16 2014