A383135 a(n) = number of iterations that n requires to reach 1 under the x -> A380891(x) map, or -1 if it never does.

0, 1, 2, 1, 3, 1, 5, 2, 3, 2, 3, 2, 4, 2, 4, 2, 6, 2, 4, 2, 6, 2, 13, 2, 13, 2, 8, 3, 8, 3, 10, 3, 10, 3, 3, 3, 10, 3, 5, 3, 10, 3, 5, 3, 5, 3, 6, 3, 5, 3, 5, 3, 17, 3, 5, 3, 5, 3, 17, 3, 3, 3, 3, 2, 12, 2, 3, 2, 5, 2, 5, 2, 12, 2, 3, 2, 7, 2, 3, 2, 7, 2, 7, 2

Offset: 1

Author

James C. McMahon and Vikram Prasad, Apr 17 2025

Keywords

• nonn

Comments

A380891(x) map is If x mod 2 = 0 then a(x) = floor(x^(1/3)) else a(x) = floor(x^(4/3)).

The map x -> A380891(x) is conjectured to eventually reach 1 for all starting x > 0. Tested for x <= 7892481.

Links

• James C. McMahon, Table of n, a(n) for n = 1..10000
• Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.

Crossrefs

Cf. A380891, A381246.

Cf. A006577.

Programs

Mathematica

fj[n_]:=If[Mod[n,2]==0,Floor[Surd[n,3]],Floor[n^(4/3)]];a383135[n_]:= Length[ NestWhileList[fj, n, # != 1 &]] - 1; Array[ a383135, 84]

Python

import sys
import gmpy2
sys.set_int_max_str_digits(0)
def floorJuggler(n) :
    a=n
    count=0
    while (a > 1) :
        b=0
        if (a%2 == 0) :
            b1=gmpy2.iroot(a,3)
            b=b1[0]
            count=count+1
        else :
            b1=gmpy2.iroot(a**4,3)
            b=b1[0]
            count=count+1
        a=b
    return count
for i in range (1, 100) :
    print (i, floorJuggler(i))