When is A111075(n) Odd?

[Paul Barry]

Seqfans,

With regard to this discussion, note that the Jacobsthal case :

Trading 
A111075 : F(n) * sum{k|n} 1/F(k), where F(k) is the kth Fibonacci number

for 

J(n) * sum{k|n} 1/J(k), where J(k) is the kth Jacobsthal number (A001045)

leads to a set of numbers a(n) that are odd precisely for n=m^2?

Paul