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

1,4

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A306251 A364027 A308967 * A233864 A133232 A137152

Adjacent sequences: A241841 A241842 A241843 * A241845 A241846 A241847

Keyword

nonn

Author

Zhi-Wei Sun, Apr 29 2014

Status

approved