A238460 -id:A238460 - cp's OEIS frontend

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

Vladimir Shevelev, Feb 26 2014

nonn

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).

			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.

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

Cf. A000040, A238460.

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

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 *)

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

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

A238501 Primes p for which x! + (p-1)!/x!==0 (mod p) has only three solutions 1<=x<=p-2.

Original entry on oeis.org

7, 11, 19, 31, 43, 47, 107, 127, 131, 151, 163, 167, 179, 191, 211, 223, 263, 283, 347, 367, 443, 487, 491, 523, 547, 587, 643, 659, 751, 827, 839, 911, 1039, 1051, 1087, 1103, 1123, 1163, 1171, 1223, 1259, 1283, 1291, 1327, 1427, 1439, 1447, 1487, 1523, 1543

Offset: 1

Vladimir Shevelev, Feb 27 2014

nonn

All terms are of the form 4*k+3.

Using Wilson's theorem, for every p>3, p==3(mod 4) we have, at least, 3 solutions in [1,p-2] of x! + (p-1)!/x!==0 (mod p): x = 1, x = (p-1)/2, x = p-2.

Cf. A238444, A238460.

kmax = 400; Select[Select[4*Range[kmax]+3, PrimeQ], (r = Range[#-2]; Count[r!+(#-1)!/r!, k_ /; Divisible[k, #]] == 3)&] (* Jean-François Alcover, Mar 05 2014 *)

a(n) is prime(k(n)) for which A238444(k(n)) = 3.

More terms from Peter J. C. Moses, Feb 27 2014

cp's OEIS Frontend

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).

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