A210144 - OEIS

%I #40 Feb 25 2024 13:12:09

%S 2,3,5,11,11,23,29,37,37,41,47,47,47,47,47,73,131,131,131,131,131,151,

%T 151,151,151,199,223,223,271,271,271,281,281,281,281,281,281,281,281,

%U 281,353,353,457,457,457,457,457,457,457,457,457,641,641,641,641,641,643,643,643,643

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

%C Conjecture: all the terms are primes and a(n)<n^2 for all n > 1.

%H Jason Yuen, <a href="/A210144/b210144.txt">Table of n, a(n) for n = 1..10000</a> (first 1172 terms from Zhi-Wei Sun)

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;df748f41.1202">A function taking only prime values</a>, a message to Number Theory List, Feb. 21, 2012.

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.

%e 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.

%t R[n_,m_]:=Union[Table[Mod[Product[Prime[j],{j,1,k}],m],{k,1,n}]]

%t Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,2,Max[2,n^2]}];

%t Print[n];Label[aa];Continue,{n,1,1000}]

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

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Mar 17 2012