A210393 a(n) = least integer m>1 such that S_k! for k=1,...,n are pairwise distinct modulo m where S_k is the sum of the first k primes.

2, 3, 7, 13, 19, 29, 43, 61, 79, 101, 131, 167, 199, 239, 293, 331, 389, 443, 503, 571, 641, 719, 797, 877, 971, 1063, 1163, 1277, 1373, 1481, 1601, 1721, 1861, 1997, 2131, 2281, 2437, 2591, 2753, 2927, 3089, 3271, 3457, 3659, 3847, 4049, 4231, 4441, 4663, 4889

Offset: 1

Author

Zhi-Wei Sun, Mar 20 2012

Keywords

• nonn

Comments

When n>1, we have S_n!=S_{n-1}!=0 (mod m) for all m=1,...,S_{n-1} and hence a(n)>S_{n-1}. Zhi-Wei Sun conjectured that a(n) is always a prime not exceeding S_n.

Examples

			a(3)=7 since 2!,(2+3)!,(2+3+5)! are pairwise distinct modulo m=7 but not pairwise distinct modulo m=2,3,4,5,6.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..720
• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

Crossrefs

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

Programs

Mathematica

s[n_]:=s[n]=Sum[Prime[k],{k,1,n}]
f[n_]:=f[n]=s[n]!
R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,Max[2,s[n-1]],s[n]}];
   Print[n];Label[aa];Continue,{n,1,720}]