A208494 Least integer m>1 such that those k! mod m with k=1,...,n are pairwise distinct.

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

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