|
|
A238532
|
|
Number of distinct factorial numbers congruent to -1 (mod n).
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
EXAMPLE
|
There are two 6's in the 7th row of A062169. Therefore a(7)=2.
|
|
MAPLE
|
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 ;
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|