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Let p_i = the i-th prime. a(i) is the smallest n>1 such that p_i divides n!-1.
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%I #24 Jun 06 2017 17:42:25

%S 3,5,9,11,5,17,4,10,15,35,39,41,45,15,18,42,48,35,17,77,41,21,43,99,

%T 96,53,22,111,125,129,120,69,25,75,155,161,83,171,177,179,189,90,195,

%U 81,105,111,82,227,101,28,239,125,255,261,267,135,236,279,141,291

%N Let p_i = the i-th prime. a(i) is the smallest n>1 such that p_i divides n!-1.

%C Since p divides (p-2)!-1, the i-th term a(i) cannot be much larger than i log i.

%H Alois P. Heinz, <a href="/A270780/b270780.txt">Table of n, a(n) for n = 3..5000</a>

%e For i=3, the third prime is 5, and 5 divides 3!-1.

%e The 7th prime is 17, and 17 divides 5!-1, so a(7)=5.

%p a:= proc(n) local k, p; p:=ithprime(n);

%p for k from 2 do if irem(k!, p)=1 then return k fi od

%p end:

%p seq(a(n), n=3..100); # _Alois P. Heinz_, Mar 23 2016

%t snpd[p_]:=Module[{n=2},While[!Divisible[n!-1,p],n++];n]; Table[snpd[p],{p,Prime[Range[3,70]]}] (* _Harvey P. Dale_, Jun 06 2017 *)

%Y Cf. A000040, A002982, A270838.

%K nonn,look

%O 3,1

%A _Peter Shor_, Mar 22 2016

%E More terms from _Alois P. Heinz_, Mar 23 2016