A210144 a(n) = least integer m>1 such that the product of the first k primes for k=1,...,n are pairwise distinct modulo m.

2, 3, 5, 11, 11, 23, 29, 37, 37, 41, 47, 47, 47, 47, 47, 73, 131, 131, 131, 131, 131, 151, 151, 151, 151, 199, 223, 223, 271, 271, 271, 281, 281, 281, 281, 281, 281, 281, 281, 281, 353, 353, 457, 457, 457, 457, 457, 457, 457, 457, 457, 641, 641, 641, 641, 641, 643, 643, 643, 643

Offset: 1

Author

Zhi-Wei Sun, Mar 17 2012

Keywords

• nonn

Comments

Conjecture: all the terms are primes and a(n) 1.

Examples

			a(3)=5 because 2, 2*3=6, 2*3*5=30 are distinct modulo m=5 but not distinct modulo m=2,3,4.
		

Links

• Jason Yuen, Table of n, a(n) for n = 1..10000 (first 1172 terms from Zhi-Wei Sun)
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012
• 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.

Crossrefs

Cf. A000040, A207982, A208494, A208643, A214196.

Programs

Mathematica

R[n_,m_]:=Union[Table[Mod[Product[Prime[j],{j,1,k}],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,2,Max[2,n^2]}];
Print[n];Label[aa];Continue,{n,1,1000}]