A238532 - OEIS

%I #39 May 24 2021 23:31:46

%S 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,

%T 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,

%U 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

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

%C Number of solutions to k! == -1 (mod n), k>=1.

%C Counts the frequency of the value n-1 in the n-th row of triangle A062169.

%C 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

%H Antti Karttunen, <a href="/A238532/b238532.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

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

%p A238532 := proc(n)

%p     local a,k ;

%p     a := 0 ;

%p     for k from 1 to n-1 do

%p         if modp(k!,n) = modp(-1,n) then

%p             a := a+1 ;

%p         end if;

%p     end do:

%p     a ;

%p end proc: # _R. J. Mathar_, Apr 02 2014

%o (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

%o (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

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

%K nonn

%O 1,7

%A _R. J. Mathar_, Apr 02 2014