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A241844 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that L(k) + m is prime, where L(k) is the k-th Lucas number.

%I A241844 #7 Apr 30 2014 04:23:18
%S A241844 0,1,1,2,1,2,3,1,1,3,3,3,2,3,2,4,3,4,4,2,2,5,4,4,4,4,2,5,4,6,5,3,1,5,
%T A241844 7,4,5,3,5,6,8,5,3,6,5,6,5,3,7,7,2,5,7,8,5,2,4,6,5,6,8,3,4,7,6,8,8,4,
%U A241844 5,5,2,5,6,4,4,5,4,12,6,6
%N A241844 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that L(k) + m is prime, where L(k) is the k-th Lucas number.
%C A241844 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.
%C A241844 (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.
%C A241844 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.
%H A241844 Zhi-Wei Sun, <a href="/A241844/b241844.txt">Table of n, a(n) for n = 1..10000</a>
%e A241844 a(5) = 1 since 5 = 1 + 4 with L(1) + 4 = 1 + 4 = 5 prime.
%e A241844 a(8) = 1 since 8 = 4 + 4 with L(4) + 4 = 7 + 4 = 11 prime.
%e A241844 a(9) = 1 since 9 = 7 + 2 with L(7) + 2 = 29 + 2 = 31 prime.
%e A241844 a(33) = 1 since 33 = 13 + 20 with L(13) + 20 = 521 + 20 = 541 prime.
%t A241844 a[n_]:=Sum[If[PrimeQ[LucasL[k]+n-k],1,0],{k,1,n-1}]
%t A241844 Table[a[n],{n,1,80}]
%Y A241844 Cf. A000032, A000040, A000045, A000204, A231201.
%K A241844 nonn
%O A241844 1,4
%A A241844 _Zhi-Wei Sun_, Apr 29 2014