
COMMENTS

Melfi proves that this sequence is infinite. The first few practical Fibonacci numbers have indices that are themselves practical (analogous to the property that the prime Fibonacci numbers have prime indices) but Melfi observes that this property is not true in general: F444 is practical although 444 itself is not.
The indices of these Fibonacci numbers are 1 (and 2), 3, 6, 12, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 126, 132, 144, 150, 156, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 270, 276, 288, 294, 300, 312, 324, 330, 336, 348, 360, 378, 384, 390, 396, 408, 420, 432, 444, ...  Amiram Eldar, May 29 2017


PROG

(PARI) is_A005153(n)=if(n%2, return(n==1)); my(P=1, f=factor(n)); for(i=2, #f~, if(f[i, 1]>1+(P*=sigma(f[i1, 1]^f[i1, 2])), return(0))); n>0
print1(1); forstep(n=3, 200, 3, if(is_A005153(t=fibonacci(n)), print1(", "t))) \\ Charles R Greathouse IV, Oct 06 2013
