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A233359 a(n) = |{0 < k < n: L(k) + q(n-k) is prime}|, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

%I A233359 #8 Dec 08 2013 09:31:34
%S A233359 0,1,1,2,3,1,2,4,2,2,3,3,2,4,3,5,1,4,5,3,1,3,3,7,3,3,4,5,2,2,9,2,4,4,
%T A233359 9,2,6,6,6,3,3,1,5,7,4,4,5,7,4,9,5,6,4,1,5,6,11,9,4,2,5,5,4,6,8,9,12,
%U A233359 3,7,5,4,10,6,7,6,3,5,8,4,4,4,4,7,7,5,1,4,9,7,4,8,7,6,5,2,3,7,11,5,5
%N A233359 a(n) = |{0 < k < n: L(k) + q(n-k) is prime}|, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).
%C A233359 Conjecture: a(n) > 0 for all n > 1.
%C A233359 We have verified this for n up to 60000.
%C A233359 Note that for n = 19976 there is no k = 0,...,n such that F(k) + q(n-k) is prime, where F(0), F(1), ... are the Fibonacci numbers.
%H A233359 Zhi-Wei Sun, <a href="/A233359/b233359.txt">Table of n, a(n) for n = 1..5000</a>
%e A233359 a(7) = 2 since L(1) + q(6) = 1 + 4 = 5 and L(6) + q(1) = 18 + 1 = 19 are both prime.
%e A233359 a(17) = 1 since L(13) + q(4) = 521 + 2 = 523 is prime.
%e A233359 a(21) = 1 since L(5) + q(16) = 11 + 32 = 43 is prime.
%e A233359 a(42) = 1 since L(22) + q(20) = 39603 + 64 = 39667 is prime.
%e A233359 a(54) = 1 since L(8) + q(46) = 47 + 2304 = 2351 is prime.
%e A233359 a(86) = 1 since L(67) + q(19) = 100501350283429 + 54 = 100501350283483 is prime.
%t A233359 a[n_]:=Sum[If[PrimeQ[LucasL[k]+PartitionsQ[n-k]],1,0],{k,1,n-1}]
%t A233359 Table[a[n],{n,1,100}]
%Y A233359 Cf. A000009, A000032, A000040, A000204, A233307, A233346, A233360.
%K A233359 nonn
%O A233359 1,4
%A A233359 _Zhi-Wei Sun_, Dec 08 2013