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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).

%I A304333 #32 May 15 2018 08:39:27
%S A304333 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,
%T A304333 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,
%U A304333 6,6,4,5,6,4,4,6,7,6,3,7,6,6,4,6,5,7,5,6,7,8
%N 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).
%C A304333 Conjecture: a(n) > 0 for all n > 1.
%C A304333 This has been verified for n up to 5*10^9.
%C A304333 See also A304331 for a similar conjecture involving Fibonacci numbers.
%C A304333 For all n, a(n) <= A130241(n). - _Antti Karttunen_, May 13 2018
%H A304333 Antti Karttunen, <a href="/A304333/b304333.txt">Table of n, a(n) for n = 1..65537</a>
%H A304333 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
%H A304333 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e A304333 a(2) = 1 with 2 - L(1) = 1 squarefree.
%e A304333 a(3) = 1 with 3 - L(1) = 2 squarefree.
%e A304333 a(67) = 2 with 67 - L(1) = 2*3*11 and 67 - L(7) = 2*19 both squarefree.
%p A304333 a := proc(n) local count, lucas, newcas;
%p A304333 count := 0; lucas := 1; newcas := 2;
%p A304333 while lucas < n do
%p A304333     if numtheory:-issqrfree(n - lucas) then count := count + 1 fi;
%p A304333     lucas, newcas := lucas + newcas, lucas;
%p A304333 od;
%p A304333 count end:
%p A304333 seq(a(n), n=1..90); # _Peter Luschny_, May 15 2018
%t A304333 f[n_]:=f[n]=LucasL[n];
%t A304333 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]
%o A304333 (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
%Y A304333 Cf. A000032, A000204, A005117, A102460, A130241, A304034, A304081, A304331.
%K A304333 nonn
%O A304333 1,4
%A A304333 _Zhi-Wei Sun_, May 11 2018