cp's OEIS Frontend

Assuming that n is not a multiple of 10, this sequence measures the difference between the number of stable digits in n^^n and the product of n times the constant congruence speed of n (see A373387 and A317905).

A negative value of a(n) means that, on average, each iteration n^^b --> n^^(b+1), with b < A372490(n), contributes fewer stable digits than what will be contributed per step once the congruence speed of n reaches its constant value.

Positive values also imply the existence of a pre-period for the given n, and indicate that its average contribution per step exceeds the constant congruence speed of n.

If n == 5 (mod 10) and n <> 5, the pre-period of the congruence speed always has length 2 (i.e., A372490(n) = 3). However, the number of stable digits observed up to that point follows two distinct rules: if n = 20*k + 5 (for positive integer k), then a(n) = (n + 1)*A373387(n); if n = 20*(k - 1) + 15, then it is n*A373387(n) + 1. The resulting residue is A373387(n) in the former case, and 1 in the latter. For n = 5, the pre-period has length 3 (and this is the only such case for n ending in 5).