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A238444 a(n) is the number of (not necessary distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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).

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, Feb 27 2014 *)

PROG

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

CROSSREFS

Cf. A000040, A238460.

Sequence in context: A046824 A130156 A139169 * A076742 A308284 A036465

Adjacent sequences:  A238441 A238442 A238443 * A238445 A238446 A238447

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Feb 26 2014

EXTENSIONS

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

STATUS

approved

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Last modified October 17 15:35 EDT 2021. Contains 348064 sequences. (Running on oeis4.)