cp's OEIS Frontend

A316141 Number of ordered pairs (k, m) with k > 1 and m > 0 such that n - F(k)*L(m) is an odd prime, where F(k) is the k-th Fibonacci number (A000045) and L(m) is the m-th Lucas number (A000204).

%I A316141 #18 Jul 12 2018 09:05:38
%S A316141 0,0,0,1,1,3,2,4,3,4,4,4,4,6,5,6,4,6,6,7,6,6,5,7,9,8,7,6,5,4,8,9,5,8,
%T A316141 7,5,9,8,8,9,7,7,8,10,8,9,6,6,9,10,8,11,6,6,10,7,6,11,8,7,11,11,9,9,
%U A316141 11,9,9,10,8,9,8,8,9,12,11,11,8,7,10,9,10,11,8,7,9,10,10,8,8,6
%N A316141 Number of ordered pairs (k, m) with k > 1 and m > 0 such that n - F(k)*L(m) is an odd prime, where F(k) is the k-th Fibonacci number (A000045) and L(m) is the m-th Lucas number (A000204).
%C A316141 Conjecture: a(n) > 0 for all n > 3. In other words, any integer n > 3 can be written as p + F(k)*L(m), where p is an odd prime, and k and m are positive integers.
%C A316141 This has been verified for all n = 4..5*10^9.
%C A316141 Note that 1623412692 is the first value of n > 3 which cannot be written as p + F(k)*L(m) with p an odd prime and k and m positive integers of the same parity.
%H A316141 Zhi-Wei Sun, <a href="/A316141/b316141.txt">Table of n, a(n) for n = 1..100000</a>
%H A316141 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e A316141 a(4) = 1 with 4 - F(2)*L(1) = 4 - 1*1 = 3 an odd prime.
%e A316141 a(5) = 1 with 5 - F(3)*L(1) = 5 - 2*1 = 3 an odd prime.
%e A316141 a(7) = 2 with 7 - F(3)*L(1) = 7 - 2*1 = 5 and 7 - F(2)*L(3) = 7 - 1*4 = 3 odd primes.
%t A316141 F[n_]:=F[n]=Fibonacci[n];
%t A316141 L[n_]:=L[n]=LucasL[n];
%t A316141 PQ[n_]:=PQ[n]=n>2&&PrimeQ[n];
%t A316141 tab={};Do[r=0;k=2;Label[aa];If[F[k]>=n,Goto[cc]];m=1;Label[bb];If[L[m]>=n/F[k],k=k+1;Goto[aa]];If[PQ[n-F[k]L[m]],r=r+1];m=m+1;Goto[bb];Label[cc];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A316141 Cf. A000040, A000045, A000204, A065091.
%K A316141 nonn
%O A316141 1,6
%A A316141 _Zhi-Wei Sun_, Jun 25 2018