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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A098666 A306251 A308967 * A233864 A133232 A137152

Adjacent sequences:  A241841 A241842 A241843 * A241845 A241846 A241847

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 29 2014

STATUS

approved

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Last modified August 13 00:27 EDT 2020. Contains 336441 sequences. (Running on oeis4.)