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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).
2
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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 25 2018
STATUS
approved