OFFSET
1,4
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
This has been verified for n up to 5*10^9.
See also A304331 for a similar conjecture involving Fibonacci numbers.
For all n, a(n) <= A130241(n). - Antti Karttunen, May 13 2018
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
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 - L(1) = 1 squarefree.
a(3) = 1 with 3 - L(1) = 2 squarefree.
a(67) = 2 with 67 - L(1) = 2*3*11 and 67 - L(7) = 2*19 both squarefree.
MAPLE
a := proc(n) local count, lucas, newcas;
count := 0; lucas := 1; newcas := 2;
while lucas < n do
if numtheory:-issqrfree(n - lucas) then count := count + 1 fi;
lucas, newcas := lucas + newcas, lucas;
od;
count end:
seq(a(n), n=1..90); # Peter Luschny, May 15 2018
MATHEMATICA
f[n_]:=f[n]=LucasL[n];
tab={}; Do[r=0; k=1; 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]
PROG
(PARI) A304333(n) = { my(u1=1, u2=3, old_u1, c=0); if(n<=2, n-1, while(u1<n, c+=issquarefree(n-u1); old_u1=u1; u1=u2; u2=old_u1+u2); c); }; \\ Antti Karttunen, May 13 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 11 2018
STATUS
approved