Lucas entry points: s(n) = smallest m >= 0 such that the n-th prime divides Lucas(m), or -1 if there is no m.
0, 2, -1, 4, 5, -1, -1, 9, 12, 7, 15, -1, 10, 22, 8, -1, 29, -1, 34, 35, -1, 39, 42, -1, -1, 25, 52, 18, -1, -1, 64, 65, -1, 23, -1, 25, -1, 82, 84, -1, 89, 45, 95, -1, -1, 11, 21, 112, 114, 57, -1, 119, 60, 125, -1, 44, -1, 135, -1, 14, 142, -1, 22, 155, -1, -1
1
Sequence A053028 contains the n such that s(n) = -1.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
(Mma) nn = 500; luc = Table[LucasL[k], {k, 0, nn}]; Table[k = 1; While[Mod[luc[[k]], Prime[n]] > 0 && k < nn, k++]; If[k == nn, -1, k - 1], {n, nn/5}]
Cf. A000032 (Lucas numbers), A001602 (Fibonacci entry points), A053028.
nonn,nice
T. D. Noe, Sep 24 2015