A304945 Number of nonnegative integers k such that n - k*L(k) is positive and squarefree, where L(k) denotes the k-th Lucas number A000032(k).

1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 1, 4, 3, 4, 2, 3, 4, 5, 2, 2, 4, 4, 2, 4, 4, 5, 2, 3, 2, 5, 2, 3, 2, 3, 2, 3, 3, 2, 2, 4, 5, 5, 2, 4, 4, 4, 1, 5, 4, 5, 3, 4, 5, 5, 3, 3, 5, 3, 2, 4, 4, 5, 2, 3, 2, 5, 3, 5, 5, 3, 3, 5, 4, 3, 3, 4, 5, 5, 2, 5, 4, 3, 1

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 9, 10, 24, 28, 64, 100, 104, 136, 153, 172, 176, 344, 496, 856, 928, 1036, 1084, 1216, 1860.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000

Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.

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(1) = 1 since 1 = 0*L(0) + 1 with 1 squarefree.
a(10) = 1 since 10 = 0*L(0) + 2*5 with 2*5 squarefree.
a(136) = 1 since 136 = 2*L(2) + 2*5*13 with 2*5*13 squarefree.
a(344) = 1 since 344 = 7*L(7) + 3*47 with 3*47 squarefree.
a(1036) = 1 since 1036 = 2*L(2) + 2*5*103 with 2*5*103 squarefree.
a(1860) = 1 since 1860 = 7*L(7) + 1657 with 1657 squarefree.

Mathematica

f[n_]:=f[n]=n*LucasL[n];
QQ[n_]:=QQ[n]=SquareFreeQ[n];
tab={}; Do[r=0; k=0; Label[bb]; If[f[k]>=n, Goto[aa]]; If[QQ[n-f[k]], r=r+1]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000032, A005117, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720, A304721, A304943.

Sequence in context: A085263 A115092 A172281 * A176298 A259575 A370062

Adjacent sequences: A304942 A304943 A304944 * A304946 A304947 A304948

Keyword

nonn

Author

Zhi-Wei Sun, May 22 2018

Status

approved