[ATH]

I looked at the LL-sequence without the mod step:
S₀=4, S₁=14, S₂=194, S₃=37634, S₄=1416317954, S₅=2005956546822746114,....

For every p where M_p is a Mersenne prime, S_p-2 is divisible by M_p and by 2 (S_p-2 always even). I found that if you divide S_p-2 by 2*M_p and add 1 to the remaining factor that number is divisible by 2^p and the remaining factor is odd. That is:

S_p-2 = 2*M_p*(n*2^p - 1)

Where n is the remaining (big) odd factor.

I only tested this for S₃, S₅, S₁₁, S₁₅, S₁₇, S₂₉ corresponding to M5, M7, M13, M17, M19, M31, so this might be the Law of small numbers . (Allthough S₂₉ isn't exactly "small" at 307,062,002 digits )

Next possible number to test this on is S₅₉ corresponding to M61, but at 3.3*10¹⁷ digits that isn't likely.