

A304331


Number of integers k > 1 such that n  F(k) is a positive squarefree number, where F(k) denotes the kth Fibonacci number A000045(k).


9



0, 1, 2, 3, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 5, 5, 2, 5, 4, 4, 2, 5, 5, 6, 3, 4, 5, 4, 2, 3, 5, 5, 2, 6, 6, 7, 4, 5, 6, 6, 4, 6, 6, 7, 4, 4, 6, 5, 4, 4, 5, 4, 2, 5, 5, 7, 3, 5, 5, 8, 4, 5, 6, 6, 4, 5, 6, 7, 5, 6, 5, 8, 4, 7, 6, 6, 4, 6, 6, 6, 5, 5, 4, 5, 5, 6, 7, 6, 4, 8
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341353, Springer, New York, 2010.
ZhiWei 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. 279310. (See also arXiv:1211.1588 [math.NT], 20122017.)


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[nf[k]], r=r+1]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]


CROSSREFS

Cf. A000045, A005117, A304034, A304081, A304333.
Sequence in context: A128630 A273040 A319982 * A241834 A086389 A128622
Adjacent sequences: A304328 A304329 A304330 * A304332 A304333 A304334


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 11 2018


STATUS

approved



