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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
Sequence in context: A043550 A237988 A291102 * A036060 A227300 A065383
KEYWORD
nonn,nice
AUTHOR
Zhi-Wei Sun, Feb 27 2012
STATUS
approved

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Last modified April 20 02:14 EDT 2024. Contains 371798 sequences. (Running on oeis4.)