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..225
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
EXAMPLE
We have a(3)=7, since m=7 divides none of 2!+(2+3)!,2!+(2+3+5)!,(2+3)!+(2+3+5)! but this fails for m=2,3,4,5,6.
MATHEMATICA
s[n_]:=s[n]=Sum[Prime[k], {k, 1, n}]
f[n_]:=s[n]!
R[n_, m_]:=Product[If[Mod[f[k]+f[j], m]==0, 0, 1], {k, 2, n}, {j, 1, k-1}]
Do[Do[If[R[n, m]==1, Print[n, " ", m]; Goto[aa]], {m, Max[2, s[n-1]], s[n]}];
Print[n]; Label[aa]; Continue, {n, 1, 225}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 20 2012
STATUS
approved