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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). 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 16:20 EDT 2021. Contains 348215 sequences. (Running on oeis4.)