A210394 a(n) = least integer m>1 such that m divides none of S_i!+S_j! with 0<i<j<=n where S_k is the sum of the first k primes.

2, 3, 7, 11, 19, 31, 43, 67, 79, 101, 131, 163, 199, 241, 283, 331, 383, 443, 503, 571, 641, 719, 797, 877, 967, 1061, 1163, 1277, 1373, 1481, 1597, 1721, 1871, 1999, 2129, 2281, 2437, 2593, 2749, 2927, 3089, 3271, 3457, 3643, 3833, 4057, 4229, 4441, 4673, 4889

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

Cf. A000040, A210393, A210186, A210144, A208494, A208643, A207982.

Sequence in context: A062576 A238686 A079739 * A211203 A350402 A158709

Adjacent sequences: A210391 A210392 A210393 * A210395 A210396 A210397

Keyword

nonn

Author

Zhi-Wei Sun, Mar 20 2012

Status

approved