A210393 - OEIS

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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

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OFFSET

1,1

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.

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.

EXAMPLE

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.

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}]

CROSSREFS