A210642 a(n) = least integer m > 1 such that k! == n! (mod m) for no 0 < k < n.

2, 2, 3, 4, 5, 9, 7, 13, 17, 17, 11, 13, 13, 19, 23, 17, 17, 29, 19, 23, 31, 31, 23, 41, 31, 29, 31, 37, 29, 31, 31, 37, 41, 41, 59, 37, 37, 59, 43, 41, 41, 59, 43, 67, 53, 53, 47, 53, 67, 59, 61, 53, 53, 79, 59, 59, 67, 73, 59, 67

Offset: 1

Author

Zhi-Wei Sun, Mar 26 2012

Keywords

• nonn

Comments

Conjecture: a(n) is a prime not exceeding 2n with the only exceptions a(4)=4 and a(6)=9.

Note that a(n) is at least n and there is at least a prime in the interval [n,2n] by the Bertrand Postulate first confirmed by Chebyshev.

Compare this sequence with A208494.

Examples

			We have a(4)=4, because 4 divides none of 4!-1!=23, 4!-2!=22, 4!-3!=18, and both 2 and 3 divide 4!-3!=18.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2500
• Oliver Gerard, Re: A new conjecture on primes, a message to Number Theory List, March 23, 2012.
• Zhi-Wei Sun, A new conjecture on primes, a message to Number Theory List, March 20, 2012.
• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Crossrefs

Cf. A000040, A208494, A210640, A210393, A210394, A210186, A210144, A208643, A207982.

Programs

Mathematica

R[n_,m_]:=If[n==1,1,Product[If[Mod[n!-k!, m]==0, 0, 1], {k, 1, n-1}]] Do[Do[If[R[n,m]==1, Print[n, " ", m]; Goto[aa]], {m,Max[2,n],2n}]; Print[n]; Label[aa]; Continue,{n,1,2500}]