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.

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, 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, 5, 5, 2, 5, 6, 4, 4, 5, 4, 12, 6, 6

Offset: 1

Author

Zhi-Wei Sun, Apr 29 2014

Keywords

• nonn

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.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000032, A000040, A000045, A000204, A231201.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[LucasL[k]+n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]