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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). 9
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified November 29 07:03 EST 2023. Contains 367429 sequences. (Running on oeis4.)