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
Keywords
Examples
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.
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.)
Programs
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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]
Comments