Least prime dividing NL(n) = sum(k=1..n) 10^(k!-1).

11, 3, 7, 43, 3, 17, 113, 3, 59, 7, 3, 11, 43, 3, 457, 2471633, 3, 71, 47711, 3, 13, 193, 3, 7, 151, 3, 461, 164821, 3, 31, 7, 3

2

The numbers NL(n) grow very quickly; the n-th term is n! digits long! The first missing term here is for NL(34), which is larger that the largest known prime. It is quite possible that NL(2) is the only prime number.

T. D. Noe, Plot of terms 2..33

Carlos B. Rivera, Prime Puzzle 825: Primes of type Liouville

(Mma) Table[p = 3; While[Mod[Sum[PowerMod[10, k! - 1, p], {k, n}], p] > 0 && p < 10000000, p = NextPrime[p]]; p, {n, 2, 33}]

Cf. A012245 (Liouvillie’s constant), S000861.

nonn,more

T. D. Noe, Apr 04 2016