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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<n, c+=issquarefree(n-u1); old_u1=u1; u1=u2; u2=old_u1+u2); c); }; \\ Antti Karttunen, May 13 2018

CROSSREFS

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

Sequence in context: A100198 A164996 A216195 * A136510 A080071 A202472

Adjacent sequences:  A304330 A304331 A304332 * A304334 A304335 A304336

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 11 2018

STATUS

approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)