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

1,4

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 A340862

Adjacent sequences: A304330 A304331 A304332 * A304334 A304335 A304336

Keyword

nonn

Author

Zhi-Wei Sun, May 11 2018

Status

approved