cp's OEIS Frontend

The present sequence is a subsequence of A068407, but it is not a subsequence of A379906 (e.g., a(4) is not a term of A379906).

Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases of d digits whose congruence speed does not stabilize in less than d + 3 iterations (e.g., the congruence speed of 807, a 3-digit number, becomes constant only at height).

As a nontrivial example, the congruence speed of a(10) := 712222747129609220545115161423787862411847003581666295807 (a 57-digit number whose constant congruence speed is also 57) becomes stable at height 60, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(712222747129609220545115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.

The smallest integer of d digits whose constant congruence speed is not yet constant at height d + 3 is 435525708925199660525680385844696084258785712222747129609220545115161423787862411847003581666295807 (a 99-digit number whose congruence speed stabilizes at height 104 to its constant value of 101).

For any n >= 2, terms of this sequence derive from one digit 5 that appears in any of the two 10-adic solutions (- {5^2^k}_oo + {2^5^k}_oo) := ...2411847003581666295807 and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943 of the fundamental 10-adic equation y^5 = y (see "The congruence speed formula" in Links). The only other candidate terms can arise from the remaining two symmetric 10-adic solutions ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 and ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193 of y^5 = y as particular patterns of 0s and 5 may occur in the corresponding (neverending) strings (e.g., '50...0').

Consequently, if n > 1 is given, a(n) is always congruent modulo 50 to 7 or 3.

Constant congruence speed of (10^n + 1)^n, i.e., a(n) = A373387((10^n + 1)^n).

Since 10^n + 1 is never a perfect power by Catalan's conjecture (Mihăilescu's theorem), it follows that if 10 does not divide n, then (10^n + 1)^n is exactly an n-th perfect power with a constant congruence speed of a(n) = n.

Moreover, for any positive integer n, the congruence speed of (10^n + 1)^n equals 2*a(n) at height 1 and then becomes stable at height 2.

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).