A304522 Number of ordered ways to write n as the sum of a Fibonacci number and a positive odd squarefree number.

1, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 3, 3, 4, 2, 4, 3, 4, 3, 4, 3, 5, 2, 4, 1, 3, 2, 2, 3, 4, 2, 5, 3, 5, 4, 4, 4, 4, 4, 5, 3, 5, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 4, 6, 3, 5, 3, 6, 3, 5, 3, 4, 3, 4, 4, 5, 4, 5, 3, 6, 4, 6, 3, 4, 3, 5, 3, 4, 3, 4, 1, 4, 4, 5, 4, 5, 3, 7

Offset: 1

Author

Zhi-Wei Sun, May 13 2018

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 27, 83, 31509.

This conjecture implies that any integer n > 1 not equal to 83 can be written as the sum of a positive Fibonacci number and a positive odd squarefree number, which has been verified for n up to 10^10. Note that 83 = 0 + 83 = 1 + 2*41, where 0 and 1 are Fibonacci numbers, and 83 and 2*41 are squarefree.

The author would like to offer 1000 US dollars as the prize for the first complete solution to his conjecture that any positive integer is the sum of a Fibonacci number and a positive odd squarefree number.

See also A304331, A304333 and A304523 for similar conjectures.

Examples

			a(1) = 1 since 1 = 0 + 1 with 0 a Fibonacci number and 1 odd and squarefree.
a(2) = 1 since 2 = 1 + 1 with 1 = A000045(1) = A000045(2) a Fibonacci number and 1 odd and squarefree.
a(27) = 1 since 27 = 8 + 19 with 8 = A000045(6) a Fibonacci number and 19 odd and squarefree.
a(83) = 1 since 83 = 0 + 83 with 0 = A000045(0) a Fibonacci number and 83 odd and squarefree.
a(31509) = 1 since 31509 = 10946 + 20563 with 10946 = A000045(21) a Fibonacci number and 20563 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: 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.)
• Index entries for sequences offering a monetary reward

Crossrefs

Cf. A000045, A005117, A304034, A304081, A304331, A304333, A304523.

Programs

Mathematica

f[n_]:=f[n]=Fibonacci[n];
QQ[n_]:=QQ[n]=n>0&&Mod[n,2]==1&&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+Boole[k==1];Goto[bb];Label[aa];tab=Append[tab,r],{n,1,90}];Print[tab]