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A304331 Number of integers k > 1 such that n - F(k) is a positive squarefree number, where F(k) denotes the k-th 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 (list; graph; refs; listen; history; text; internal format)
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

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

Zhi-Wei Sun, May 11 2018

STATUS

approved

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Last modified May 6 02:46 EDT 2021. Contains 343579 sequences. (Running on oeis4.)