A385613 Number of steps that n requires to reach 0 under the map: x-> x^2 - 1 if x is an odd prime, floor(x/3) if x is even, otherwise x - 1. a(n) = -1 if 0 is never reached.

0, 1, 1, 3, 2, 4, 2, 7, 2, 3, 4, 9, 3, 6, 3, 4, 5, 8, 3, 10, 3, 4, 8, 14, 3, 4, 3, 4, 4, 10, 5, 15, 5, 6, 10, 11, 4, 10, 4, 5, 7, 8, 4, 14, 4, 5, 5, 10, 6, 7, 6, 7, 9, 15, 4, 5, 4, 5, 11, 9, 4, 18, 4, 5, 5, 6, 9, 10, 9, 10, 15, 9, 4, 22, 4, 5, 5, 6, 4, 11, 4

Offset: 0

Author

Ya-Ping Lu, Jul 04 2025

Keywords

• sign

Comments

n = 122827 is the smallest starting number that ends up in a loop. The loop contains 33 elements: 122827 -> 15086471928 -> 5028823976 -> 1676274658 -> 558758219 -> 558758218 -> 186252739 -> 186252738 -> 62084246 -> 20694748 -> 6898249 -> 47585839266000 -> 15861946422000 -> 5287315474000 -> 1762438491333 -> 1762438491332 -> 587479497110 -> 195826499036 -> 65275499678 -> 21758499892 -> 7252833297 -> 7252833296 -> 2417611098 -> 805870366 -> 268623455 -> 268623454 -> 89541151 -> 89541150 -> 29847050 -> 9949016 -> 3316338 -> 1105446 -> 368482 -> 122827.

No other loop is found for n < 164901049.

Conjecture: a(n) = -1 occurs only when starting number n runs into a loop.

Examples

			a(5) = 4 because iterating the map on n = 5 results in 0 in 4 steps: 5 -> 5^2-1=24 -> floor(24/3)=8 -> floor(8/3)=2 -> floor(2/3)=0.
a(9949031) = -1 because iterating the map on n = 9949031 ends up in the 33-member loop in 5 steps: 9949031 -> 9949030 -> 3316343 -> 3316342 -> 1105447 -> 1105446.
		

Crossrefs

Cf. A339991, A340801, A384713.

Programs

Python

from sympy import isprime
def A385613(n):
    S = {n}
    while n != 0:
        n = n//3 if n%2 == 0 else n*n - 1 if isprime(n) else n - 1
        if n in S: return -1
        S.add(n)
    return len(S) - 1