A230315 -id:A230315 - cp's OEIS frontend

A238532 Number of distinct factorial numbers congruent to -1 (mod n).

0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

R. J. Mathar, Apr 02 2014

nonn

Number of solutions to k! == -1 (mod n), k>=1.
Counts the frequency of the value n-1 in the n-th row of triangle A062169.
Values 1..9 occur for the first time at n = 2, 7, 23, 59, 227, 401, 71, 3643, 62939, which are all prime numbers (see also A230315). Sequence A256519 gives composite k for which a(k) > 0. - Antti Karttunen, May 24 2021

			There are two 6's in the 7th row of A062169. Therefore a(7)=2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to factorial numbers

Cf. A000142, A049046, A062169, A230315, A256519.

A238532 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 1 to n-1 do
        if modp(k!,n) = modp(-1,n) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 02 2014

A238532(n) = { my(m=1,s=0); for(k=1,oo,m *= k; if(!(m%n),return(s), if(1+(m%n)==n, s++))); }; \\ Antti Karttunen, May 24 2021

A238532(n) = { my(m=Mod(1,n),s=0,x); for(k=1,oo, m *= Mod(k,n); x = lift(m); if(!x,return(s), if(x==(n-1), s++))); }; \\ (Much faster than above program) - Antti Karttunen, May 24 2021

cp's OEIS Frontend

A238532 Number of distinct factorial numbers congruent to -1 (mod n).

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