

A304522


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


8



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
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OFFSET

1,3


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.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..100000
ZhiWei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341353, Springer, New York, 2010.
ZhiWei 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. 279310. (See also arXiv:1211.1588 [math.NT], 20122017.)
Index entries for sequences offering a monetary reward


EXAMPLE

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.


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[nf[k]], r=r+1]; k=k+1+Boole[k==1]; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]


CROSSREFS

Cf. A000045, A005117, A304034, A304081, A304331, A304333, A304523.
Sequence in context: A237531 A238504 A031356 * A308641 A024676 A093429
Adjacent sequences: A304519 A304520 A304521 * A304523 A304524 A304525


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 13 2018


STATUS

approved



