Magic multipliers

1, 1, 3, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 9, 1, 1, 3, 1, 1, 1, 9, 3, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 30, 3, 2, 3, 39, 1, 1, 1, 6, 3, 34, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 9, 1, 1, 1, 2, 1, 37, 3, 9, 6, 1, 8, 1, 1, 2, 1, 3, 2, 10, 1, 1, 11, 19, 3, 1, 1, 1, 1, 2, 1, 1, 7, 1, 47, 3, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 8, 3, 1, 24, 1, 3, 10, 1, 1, 1, 1, 9, 3, 3, 13, 1, 6, 1, 21, 10, 9, 7, 3, 1, 1, 1, 7, 2, 3, 19, 1, 1, 1, 6, 1, 1, 2, 37, 1, 14, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 8, 3, 1, 7, 1, 3, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 40, 3, 1, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 8, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 4, 3, 1, 1, 7, 9, 1, 1, 1, 1, 60, 3, 1, 1, 1, 4, 3, 1, 2, 3, 6, 1, 3, 1, 10, 1, 1, 1, 52, 1, 1, 13, 1, 1, 3, 1, 19, 1, 1, 2, 31, 3, 1, 1, 1, 10, 3, 1, 2, 1, 1, 1, 29, 1, 1, 1, 3, 1, 1, 1, 3, 3, 110, ...

This sequence looks superficially like a simple continued-fraction expansion of some constant, but it's not. There are no numbers (I conjecture) ending with the digit 5. How did I arrive at my sequence? They are the magic multipliers (the final number on each line) in the following:

n =  1:  (37+1)/2 =  19*1
n =  2:  (73+1)/2 =  37*1
n =  3: (113+1)/2 =  19*3
n =  4: (149-1)/2 =  37*2
n =  5: (157+1)/2 =  79*1
n =  6: (193+1)/2 =  97*1
n =  7: (269-1)/2 =  67*2
n =  8: (277+1)/2 = 139*1
n =  9: (313+1)/2 = 157*1
n = 10: (353+1)/2 =  59*3
n = 11: (389-1)/2 =  97*2
n = 12: (397+1)/2 = 199*1
n = 13: (457+1)/2 = 229*1
n = 14: (557+1)/2 =  31*9

etc.

The initial number on each line is A135952(n). Note that if the magic multiplier is odd, we add 1 before dividing by 2; if it is even, we subtract 1 before dividing by 2. The number immediately after each equal sign is the prime p where A135952(n) divides composite Fibonacci(p). In my sequence of magic multipliers, the first occurrences of the positive integers are at indices:

1, 4, 3, 157, 0, 24, 91, 71, 14, 78, 81, 802, 124, 149, 0, 720, 436, 292, 82, 347, 128, 389, 598, 113, 0, 1245, 454, 1728, 270, 39, 258, ...