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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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 A169623

Adjacent sequences:  A304942 A304943 A304944 * A304946 A304947 A304948

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 22 2018

STATUS

approved

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Last modified August 15 01:56 EDT 2020. Contains 336485 sequences. (Running on oeis4.)