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

1,6

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.

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.)

Example

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.

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]

Crossrefs

Cf. A000040, A000045, A000204, A065091.

Sequence in context: A134559 A333773 A007456 * A119707 A354679 A307118

Adjacent sequences: A316138 A316139 A316140 * A316142 A316143 A316144

Keyword

nonn

Author

Zhi-Wei Sun, Jun 25 2018

Status

approved