OFFSET
1,4
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 5 can be written as k + m with k > 1 and m > 0 such that L(k) + m is prime.
(ii) Any integer n > 1 can be written as k + m with k > 0 and m > 0 such that F(k) + 2*m is prime, where F(k) is the k-th Fibonacci number given by A000045.
This is similar to the conjecture in A231201. We have verified parts (i) and (ii) for n up to 7*10^6 and 3.7*10^6 respectively.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(5) = 1 since 5 = 1 + 4 with L(1) + 4 = 1 + 4 = 5 prime.
a(8) = 1 since 8 = 4 + 4 with L(4) + 4 = 7 + 4 = 11 prime.
a(9) = 1 since 9 = 7 + 2 with L(7) + 2 = 29 + 2 = 31 prime.
a(33) = 1 since 33 = 13 + 20 with L(13) + 20 = 521 + 20 = 541 prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[LucasL[k]+n-k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 29 2014
STATUS
approved