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A304523 Number of ordered ways to write n as the sum of a Lucas number (A000032) and a positive odd squarefree number. 7
0, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 3, 1, 4, 2, 3, 2, 4, 3, 3, 3, 4, 3, 4, 3, 3, 1, 2, 1, 4, 2, 4, 3, 4, 3, 4, 3, 3, 3, 5, 3, 5, 2, 5, 2, 4, 2, 5, 2, 5, 2, 4, 2, 5, 3, 2, 3, 6, 3, 5, 3, 6, 2, 5, 2, 5, 1, 6, 3, 5, 3, 5, 3, 3, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 2, 5, 4, 5, 4, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 11, 13, 27, 29, 67, 139, 193, 247, 851.

It has been verified that a(n) > 0 for all n = 2..5*10^9.

See also A304331, A304333 and A304522 for similar conjectures.

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(3) = 1 since 3 = A000032(0) + 1 with 1 odd and squarefree.

a(27) = 1 since 27 = A000032(3) + 23 with 23 odd and squarefree.

a(29) = 1 since 29 = A000032(6) + 11 with 11 odd and squarefree.

a(67) = 1 since 67 = A000032(0) + 5*13 with 5*13 odd and squarefree.

a(247) = 1 since 247 = A000032(6) + 229 with 229 odd and squarefree.

a(851) = 1 since 851 = A000032(0) + 3*283 with 3*283 odd and squarefree.

MATHEMATICA

f[n_]:=f[n]=LucasL[n];

QQ[n_]:=QQ[n]=n>0&&Mod[n, 2]==1&&SquareFreeQ[n];

tab={}; Do[r=0; k=0; Label[bb]; If[k>0&&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, 90}]; Print[tab]

CROSSREFS

Cf. A000032, A005117, A304034, A304081, A304331, A304333, A304522.

Sequence in context: A270559 A231727 A270616 * A020649 A183024 A067131

Adjacent sequences:  A304520 A304521 A304522 * A304524 A304525 A304526

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 13 2018

STATUS

approved

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Last modified March 28 11:00 EDT 2020. Contains 333083 sequences. (Running on oeis4.)