0,4

See A373387 for the definition of "constant congruence speed" in the decimal numeral system.

As for the decimal numeral system case (see A373387), we assume that a(0) = a(1) = 1.

Let ord_2(x) and ord_3(x) indicate, respectively, the 2-adic and 3-adic valuation of x.

If n == 1,5 (mod 6), than a(n) = min(ord_2(n^2 - 1) + ord_3(n^2 - 1)); if n == 3 (mod 6), than a(n) = ord_2(n^2 - 1) - 1; if n == 2,4 (mod 6), than a(n) = ord_3(n^2 - 1); if n == 0 (mod 6), than the congruence speed of n never becomes stable (and thus, a(n) = -1).

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.

Marco Ripà and Gabriele Di Pietro, A Compact Notation for Peculiar Properties Characterizing Integer Tetration, Zenodo, 2025.

Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.

Wikipedia, Tetration.

a(n) = min(ord_2(n^2 - 1) + ord_3(n^2 - 1)) for n == 1,5 (mod 6),

a(n) = ord_2(n^2 - 1) - 1 for n == 3 (mod 6),

a(n) = ord_3(n^2 - 1) for n == 2,4 (mod 6),

a(n) = -1 for n == 0 (mod 6).

(Python)

def v2(x):

count = 0

while x % 2 == 0 and x > 0:

x //= 2

count += 1

return count

def v3(x):

count = 0

while x % 3 == 0 and x > 0:

x //= 3

count += 1

return count

def a(n):

mod_6 = n % 6

if n == 0 or n == 1:

return 0

if mod_6 == 0:

return -1

if mod_6 in {1, 5}:

return min(v2(n * n - 1) - 1, v3(n * n - 1))

elif mod_6 == 3:

return v2(n * n - 1) - 1

elif mod_6 in {2, 4}:

return v3(n * n - 1)

def generate_sequence():

seq = []

for n in range(1001):

seq.append(a(n))

return seq

sequence = generate_sequence()

print("a(0), a(1), a(2), ..., a(1000) =", ", ".join(map(str, sequence)))

Cf. A317905, A373387, A390597.

Sequence in context: A321913 A247378 A094102 * A220091 A063746 A293429

Adjacent sequences: A390595 A390596 A390597 * A390599 A390600 A390601

sign,base

Marco Ripà, Nov 12 2025

approved