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 A208494 Least integer m>1 such that those k! mod m with k=1,...,n are pairwise distinct. 12
 2, 2, 3, 7, 10, 13, 13, 13, 31, 37, 37, 37, 61, 61, 61, 83, 83, 83, 127, 127, 127, 127, 127, 179, 179, 179, 179, 179, 193, 193, 193, 193, 193, 193, 193, 193, 277, 277, 277, 277, 277, 277, 383, 383, 383, 383, 383, 479, 479, 479, 479, 479, 479, 479, 541, 541, 541, 541, 541, 541, 541, 541, 541, 641, 641, 641, 641, 641, 641, 641, 641, 641, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 877, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013, 1013 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000142, A207982. Sequence in context: A043550 A237988 A291102 * A036060 A227300 A065383 Adjacent sequences: A208491 A208492 A208493 * A208495 A208496 A208497 KEYWORD nonn,nice AUTHOR Zhi-Wei Sun, Feb 27 2012 STATUS approved

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