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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 17 2018
STATUS
approved