When is A111075(n) Odd?

[Paul D. Hanna]

Seqfans,
       In case someone has already extended Leroy's sequence A111075,
I would like to make an observation here regarding:
http://www.research.att.com/projects/OEIS?Anum=A111075
1,2,3,7,6,21,14,50
Name: F(n) * sum{k|n} 1/F(k), where F(k) is the kth Fibonacci number.
 
Here are more terms:
1,2,3,7,6,21,14,50,52,122,90,427,234,784,1038,2351,1598,
6860,4182,17262,17262,35622,28658,139703,90031,243308,
300405,766850,514230,2367006,1346270,5188658,5326470,
11409346,11782764,44717548,24157818,78185688,95140422,
(PARI) a(n)=fibonacci(n)*sumdiv(n,d,1/fibonacci(d))
 
The question that came to mind was: "when is A111075(n) odd?"
This gives a new sequence:
"n for which  A111075(n) is odd"
1,3,4,6,12,16,24,25,27,48,49,54,64,75,96,100,108,121,147,
150,169,192,196,216,243,256,289,294,300,361,363,384,400,432,
484,486,507,529,588,600,625,675,676,726,768,784,841,864,867,
961,972,1014,1024,1083,1156,1176,1200,1225,1323,1350,1369,
 
Conjecture: A111075(n) is odd whenever: 
(i)  n = m^2 for all m>=1 such that 3 does not divide m, and 
(ii)  n = 3*A028982(m) for all m>=1.
Note that A028982 lists positive integers having an odd sum of divisors.
 
Can anyone support this conjecture?  Is this a trivial observation? 
Perhaps Ralf Stephan or other Fibonacci experts can answer this question.
 
Thanks,
      Paul
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