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).

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Jun 25 2018

Keywords

• nonn

Comments

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.

This has been verified for all n = 4..5*10^9.

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.

Examples

			a(4) = 1 with 4 - F(2)*L(1) = 4 - 1*1 = 3 an odd prime.
a(5) = 1 with 5 - F(3)*L(1) = 5 - 2*1 = 3 an odd prime.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
• Zhi-Wei Sun, Conjectures on representations involving primes, 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 arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A000040, A000045, A000204, A065091.

Programs

Mathematica

F[n_]:=F[n]=Fibonacci[n];
L[n_]:=L[n]=LucasL[n];
PQ[n_]:=PQ[n]=n>2&&PrimeQ[n];
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]