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A238444 a(n) is the number of (not necessarily distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n).

%I A238444 #34 Jan 29 2023 19:42:31
%S A238444 0,1,2,3,3,2,4,3,7,4,3,2,2,3,3,4,7,8,5,9,4,5,7,4,4,2,5,3,4,2,3,3,6,5,
%T A238444 6,3,2,3,3,2,3,2,3,4,2,7,3,3,7,2,6,5,2,5,4,3,4,5,4,2,3,6,7,7,2,4,5,2,
%U A238444 3,2,2,7,3,2,5,5,6,6,10,2,5,2,5,2,5,3,6
%N A238444 a(n) is the number of (not necessarily distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n).
%C A238444 If n>1, then from Wilson's theorem (i=1) and a simple corollary to it (i=prime(n)-2) we have a(n)>=2.
%C A238444 Conjecture: There are infinitely many n for which a(n)=2 (the sequence of corresponding primes is A238460).
%H A238444 Peter J. C. Moses, <a href="/A238444/b238444.txt">Table of n, a(n) for n = 1..1000</a>
%e A238444 Let n=4, prime(n)=7. Consider integers i!+6!/i!, i=1,2,3,4,5: 721,362,126,54,126. Among them 721,126,126 are divisible by 7. So a(4)=3.
%p A238444 A238444 := proc(n)
%p A238444     local p,a,i ;
%p A238444     p := ithprime(n) ;
%p A238444     a := 0 ;
%p A238444     for i from 1 to p-2 do
%p A238444         if modp( i!+(p-1)!/i!,p)= 0 then
%p A238444             a := a+1 ;
%p A238444         end if;
%p A238444     end do;
%p A238444     a ;
%p A238444 end proc:
%p A238444 seq(A238444(n),n=1..20) ; # _R. J. Mathar_, Mar 06 2014
%t A238444 a[n_] := Module[{p, r}, p = Prime[n]; r = Range[p-2]; Count[r!+(p-1)!/r!, k_ /; Divisible[k, p]]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 27 2014 *)
%o A238444 (PARI) a(n) = sum(i=1, prime(n)-2, ((i!+(prime(n)-1)!/i!) % prime(n)) == 0); \\ _Michel Marcus_, Feb 27 2014
%Y A238444 Cf. A000040, A238460.
%K A238444 nonn
%O A238444 1,3
%A A238444 _Vladimir Shevelev_, Feb 26 2014
%E A238444 More terms from _Peter J. C. Moses_, Feb 26 2014