OFFSET

1,3

COMMENTS

Conjecture: a(n) > 0 for all n > 1. In other words, every n = 2,3,... can be written as the sum of a positive Fibonacci number and a positive squarefree number.

This has been verified for n up to 10^10.

See also A304333 for a similar conjecture involving Lucas numbers.

LINKS

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

Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.

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(2) = 1 with 2 - F(2) = 1 squarefree.

a(53) = 2 with 53 - F(3) = 3*17 and 53 - F(9) = 19 both squarefree.

MATHEMATICA

f[n_]:=f[n]=Fibonacci[n];

tab={}; Do[r=0; k=2; Label[bb]; If[f[k]>=n, Goto[aa]]; If[SquareFreeQ[n-f[k]], r=r+1]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]

CROSSREFS

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 11 2018

STATUS

approved