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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. 6
 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000040, A210394, A210186, A210144, A208494, A208643, A207982. Sequence in context: A076974 A051484 A101415 * A045331 A053613 A013645 Adjacent sequences:  A210390 A210391 A210392 * A210394 A210395 A210396 KEYWORD nonn AUTHOR Zhi-Wei Sun, Mar 20 2012 STATUS approved

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Last modified June 1 00:27 EDT 2020. Contains 334756 sequences. (Running on oeis4.)