A304689 Number of nonnegative integers k such that n - F(k)*F(k+1) is positive and squarefree, where F(k) denotes the k-th Fibonacci number A000045(k).

1, 2, 3, 2, 2, 2, 4, 3, 2, 1, 3, 3, 3, 2, 3, 4, 4, 2, 3, 3, 4, 3, 4, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 3, 4, 4, 4, 3, 4, 3, 5, 3, 5, 4, 4, 3, 4, 4, 3, 2, 2, 3, 5, 3, 3, 2, 5, 3, 5, 2, 5, 4, 4, 2, 2, 4, 4, 4, 3, 4, 5, 4, 5, 4, 4, 3, 4, 3, 5, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 1, 4, 3, 5, 3, 5, 2, 5, 3, 3, 2

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 10, 90, 66690, 67452, 22756020.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000

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

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(1) = 1 with 1 - F(0)*F(1) = 1 squarefree.
a(10) = 1 with 10 - F(0)*F(1) = 2*5 squarefree.
a(90) = 1 with 90 - F(1)*F(2) = 89 squarefree.
a(66690) = 1 with 66690 - F(10)*F(11) = 66690 - 55*89 = 5*17*727 squarefree.
a(67452) = 1 with 67452 - F(1)*F(2) = 37*1823 squarefree.
a(22756020) = 1 with 22756020 - F(2)*F(3) = 2*11378009 squarefree.

Mathematica

f[n_]:=f[n]=Fibonacci[n]*Fibonacci[n+1];
QQ[n_]:=QQ[n]=SquareFreeQ[n];
tab={}; Do[r=0; k=0; Label[bb]; If[f[k]>=n, Goto[aa]]; If[QQ[n-f[k]], r=r+1]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000045, A005117, A304034, A304081, A304331, A304333, A304522, A304523.

Sequence in context: A248973 A305048 A205717 * A288677 A187757 A286529

Adjacent sequences: A304686 A304687 A304688 * A304690 A304691 A304692

Keyword

nonn

Author

Zhi-Wei Sun, May 17 2018

Status

approved