A304523 Number of ordered ways to write n as the sum of a Lucas number (A000032) and a positive odd squarefree number.

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

Offset: 1

Author

Zhi-Wei Sun, May 13 2018

Keywords

• nonn

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.

Examples

			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.
		

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.)

Crossrefs

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

Programs

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]