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

1,1

Comments

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

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A207982 A259155 A354400 * A243357 A066159 A316794

Adjacent sequences: A210141 A210142 A210143 * A210145 A210146 A210147

Keyword

nonn

Author

Zhi-Wei Sun, Mar 17 2012

Status

approved