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).

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

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