A304522 - OEIS

%I #21 Nov 01 2018 06:38:59

%S 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,

%T 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,

%U 4,5,3,6,4,6,3,4,3,5,3,4,3,4,1,4,4,5,4,5,3,7

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

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

%C 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.

%C 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.

%C See also A304331, A304333 and A304523 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A304522/b304522.txt">Table of n, a(n) for n = 1..100000</a>

%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.

%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)

%H <a href="/index/Se#tiny4">Index entries for sequences offering a monetary reward</a>

%e a(1) = 1 since 1 = 0 + 1 with 0 a Fibonacci number and 1 odd and squarefree.

%e a(2) = 1 since 2 = 1 + 1 with 1 = A000045(1) = A000045(2) a Fibonacci number and 1 odd and squarefree.

%e a(27) = 1 since 27 = 8 + 19 with 8 = A000045(6) a Fibonacci number and 19 odd and squarefree.

%e a(83) = 1 since 83 = 0 + 83 with 0 = A000045(0) a Fibonacci number and 83 odd and squarefree.

%e a(31509) = 1 since 31509 = 10946 + 20563 with 10946 = A000045(21) a Fibonacci number and 20563 odd and squarefree.

%t f[n_]:=f[n]=Fibonacci[n];

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

%t 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]

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

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, May 13 2018