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

1,7

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to factorial numbers

Example

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

Maple

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

Prog

(PARI) 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
(PARI) 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

Crossrefs

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

Sequence in context: A292261 A035153 A069847 * A097249 A112313 A297115

Adjacent sequences: A238529 A238530 A238531 * A238533 A238534 A238535

Keyword

nonn

Author

R. J. Mathar, Apr 02 2014

Status

approved