When is A111075(n) Odd?

[Klaus Brockhaus]

Paul, Leroy, seqfans,

is there a proof that A111075(n) is an integer for all n? By a PARI program I got the following
list of numbers < 3000 for which this seemingly is not true:

{for(n=1,3000,d=divisors(n);p=fibonacci(n)*sum(j=1,length(d),1/fibonacci(d[j]));
if(denominator(p)>1,print1(n,",")))}
2135,2187,2275,2330,2345,2408,2500,2510,2523,2555,2626,2650,2665,2666,
2738,2758,2770,2782,2786,2795,2798,2806,2810,2812,2818,2822,2830,2836,
2858,2883,2905,2938,2950,2954,2956,2962,2966,2974,2975,2978,2984,2986,2998.

Could someone check the program and/or run it on a recent version of PARI ? (I use the old
version 2.0.17)

Thanks,

Klaus
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%I A111075
%S A111075 1,2,3,7,6,21,14,50,52,122,90,427,234,784,1038,2351,1598,6860,4182,
%T A111075 17262,17262,35622,28658,139703,90031,243308,300405,766850,514230,
%U A111075 2367006,1346270,5188658,5326470,11409346,11782764,44717548,24157818
%N A111075 F(n) * sum{k|n} 1/F(k), where F(k) is the kth Fibonacci number.
%C A111075 a(n) = a(n+1) for n = 20, but for no other n < 25000. - Klaus Brockhaus
%e A111075 a(6) = F(6) sum{k|6} 1/F(k) = F(6) * (1/F(1) + 1/F(2) + 1/F(3) + 1/F(6)) = 8 * (1/1
+ 1/1 + 1/2 + 1/8) = 21.
%o A111075 (PARI) {for(n=1,37,d=divisors(n);print1(fibonacci(n)*sum(j=1,length(d),
1/fibonacci(d[j])),","))}
%Y A111075 Cf. A000045.
%K A111075 nonn,new
%O A111075 1,2
%A A111075 Leroy Quet (qq-quet(AT)mindspring.com), Oct 10 2005
%E A111075 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 12 2005