OFFSET
1,1
COMMENTS
On Feb 27 2012 Zhi-Wei Sun conjectured that a(n) is a prime with the only exception a(5)=10, and that a(n) does not exceed n^2/2 for all n=2,3,4,... He also conjectured that max{n>0: 1!,...,n! are pairwise incongruent mod p} is asymptotically equivalent to sqrt(p), where p is an odd prime.
He guessed that if we replace k! in the definition of a(n) by (-1)^k*k! then a(n) is a prime with the only exception a(3)=6. If we replace k! in the definition of a(n) by (2k)! or (-1)^k*(2k)!, then Zhi-Wei Sun conjectured that a(n) will take only prime values.
He also has similar conjectures involving (r*k)! or (-1)^k*(r*k)! with r>2.
LINKS
Zhi-Wei Sun and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 400 terms from Sun)
Zhi-Wei Sun, A function taking only prime values, a message to Number Theory List, Feb 21 2012.
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
EXAMPLE
For n=5 we have a(5)=10 since 1!=1, 2!=2, 3!=6, 4!=24 and 5!=120 are pairwise incongruent mod 10 but not pairwise incongruent modulo any of 2,3,...,9.
MATHEMATICA
R[n_, i_]:=Union[Table[Mod[k!, i], {k, 1, n}]]
Do[Do[If[Length[R[n, i]]==n, Print[n, " ", i]; Goto[aa]], {i, 2, Max[n^2, 2]}];
Print[n]; Label[aa]; Continue, {n, 1, 1000}]
PROG
(PARI) has(n, m)=my(t=1); #Set(vector(n, i, t=(t*i)%m))==n
a(n, last=2)=while(!has(n, last), last++); last
t=2; vector(100, n, t=a(n, t)) \\ Charles R Greathouse IV, Jul 31 2016
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Feb 27 2012
STATUS
approved