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

0,4

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.

Links

Table of n, a(n) for n=0..80.

Example

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.

Prog

(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

Crossrefs

Cf. A339991, A340801, A384713.

Sequence in context: A322979 A106288 A013633 * A334662 A016559 A104566

Adjacent sequences: A385610 A385611 A385612 * A385614 A385615 A385616

Keyword

sign

Author

Ya-Ping Lu, Jul 04 2025

Status

approved