A304333 Number of positive integers k such that n - L(k) is a positive squarefree number, where L(k) denotes the k-th Lucas number A000204(k).

0, 1, 1, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 5, 2, 3, 4, 5, 2, 4, 4, 4, 3, 5, 4, 4, 2, 3, 3, 5, 3, 5, 5, 5, 4, 4, 5, 4, 4, 6, 5, 6, 3, 6, 4, 5, 3, 6, 5, 6, 3, 5, 4, 5, 3, 3, 4, 6, 4, 6, 4, 7, 3, 6, 4, 6, 2, 6, 6, 6, 4, 5, 6, 4, 4, 6, 7, 6, 3, 7, 6, 6, 4, 6, 5, 7, 5, 6, 7, 8

Offset: 1

Author

Zhi-Wei Sun, May 11 2018

Keywords

• nonn

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

Examples

			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.
		

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.)

Crossrefs

Cf. A000032, A000204, A005117, A102460, A130241, A304034, A304081, A304331.

Programs

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]

PARI

A304333(n) = { my(u1=1,u2=3,old_u1,c=0); if(n<=2,n-1,while(u1Antti Karttunen, May 13 2018