# [seqfan] Re: Observations on some odd Fibonacci numbers

Richard Mathar mathar at strw.leidenuniv.nl
Tue Oct 5 17:34:14 CEST 2010

```http://list.seqfan.eu/pipermail/seqfan/2010-October/006140.html

vs>    I consider the following subsequence of Fibonacci numbers:
vs> 21,55,377,987,6765,17711,121393,2178309,5702887,39088169,1836311903,...
vs> with the definition: a(n) is the n-th odd Fibonacci number F with the
vs> property: F has a proper Fibonacci divisor G>1, but F/G has not.

I think F(40)=102334155 with G=6765 is missing, where F/G=15127
has proper divisors 7 and 2161 which are both prime. The sequence
starts according to my (hereby no longer private) calculations as
n, F(n)
8, 21
10, 55
14, 377
16, 987
22, 17711
26, 121393
32, 2178309
34, 5702887
38, 39088169
40, 102334155
46, 1836311903
49, 7778742049
50, 12586269025
56, 225851433717
58, 591286729879
62, 4052739537881
64, 10610209857723
70, 190392490709135
74, 1304969544928657
80, 23416728348467685
82, 61305790721611591
86, 420196140727489673
88, 1100087778366101931
94, 19740274219868223167
98, 135301852344706746049
104, 2427893228399975082453
106, 6356306993006846248183
110, 43566776258854844738105
112, 114059301025943970552219
118, 2046711111473984623691759

Note the presense of at least one odd index n=49 above which says that F(2n)=F(n)*L(n)
does not catch all cases of interest.

RJM

# test whether n is a fibonacci number, true if yes.
isfibo := proc(n)
for i from 1 do
f := combinat[fibonacci](i) ;
if f > n then
return false;
elif f = n then
return true;
fi ;
od;
end proc:

# test if this is an odd F(n) which has a G|F , G in A000045 not equal to 1 or F,
# and no d|G with d<>1, d<> G and d in A000045.
isVSa := proc(n)
f := combinat[fibonacci](n) ;
if type(f,'even') then
return false;
fi ;
for G in (numtheory[divisors](f) minus {1,f}) do
# test all proper fibonacci divisors G of F
if isfibo(G) then
foverg := f/G ;
fghasf := false;
# test all proper fibonacci divisors of G/F
for fgd in (numtheory[divisors](foverg) minus {1,foverg}) do
if isfibo(fgd) then
fghasf := true;
break;
fi;
end do:
# now G|F but there is no d|G with d in A000045
if fghasf = false then
return true;
fi;
fi ;
od;
return false;
end proc:
for n from 1 to 80 do
if isVSa(n) then
print(n,combinat[fibonacci](n)) ;
fi;
end do:

```