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