Composite Fibonacci Numbers with Prime Index

Theorem
The sequence of composite Fibonacci numbers with a prime index begins:
 * $4181, 1 \, 346 \, 269, 24 \, 157 \, 817, 165 \, 580 \, 141, \ldots$

The corresponding sequence of prime indices begins:
 * $19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, \ldots$

Proof
By observation: