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).

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, 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, 3, 2, 2, 7, 3, 2, 5, 5, 6, 6, 10, 2, 5, 2, 5, 2, 5, 3, 6

Offset: 1

Author

Vladimir Shevelev, Feb 26 2014

Keywords

• nonn

Comments

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.

Conjecture: There are infinitely many n for which a(n)=2 (the sequence of corresponding primes is A238460).

Examples

			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.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A238460.

Programs

Maple

A238444 := proc(n)
    local p,a,i ;
    p := ithprime(n) ;
    a := 0 ;
    for i from 1 to p-2 do
        if modp( i!+(p-1)!/i!,p)= 0 then
            a := a+1 ;
        end if;
    end do;
    a ;
end proc:
seq(A238444(n),n=1..20) ; # R. J. Mathar, Mar 06 2014

Mathematica

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 *)

PARI

a(n) = sum(i=1, prime(n)-2, ((i!+(prime(n)-1)!/i!) % prime(n)) == 0); \\ Michel Marcus, Feb 27 2014

Extensions

More terms from Peter J. C. Moses, Feb 26 2014