A340420 The number of steps that n requires to reach 1 under the map: m -> m/2 if m is even, m-> 3*m + 1 if m is an odd prime, otherwise m -> m - 1. a(n) = -1 if 1 is never reached.

0, 1, 7, 2, 5, 8, 16, 3, 4, 6, 14, 9, 9, 17, 18, 4, 12, 5, 20, 7, 8, 15, 16, 10, 11, 10, 11, 18, 18, 19, 19, 5, 6, 13, 14, 6, 21, 21, 22, 8, 22, 9, 9, 16, 17, 17, 17, 11, 12, 12, 13, 11, 11, 12, 13, 19, 20, 19, 32, 20, 20, 20, 21, 6, 7, 7, 27, 14, 15, 15, 15

Offset: 1

Author

Ya-Ping Lu, Jan 07 2021

Keywords

• nonn
• look

Comments

Conjecture: a(n) is never equal to -1.

Examples

			a(3) = 7 because 3*3 + 1 = 10 -> 10/2 = 5 -> 3*5 + 1 = 16 -> 16/2 = 8 -> 8/2 = 4 -> 4/2 = 2 -> 2/2 = 1 -> 1.
a(14) = 17 because 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.
The 39 terms for a(n) <= 9 are given in the figure below.
145 288 12 42 13 80 133 264 260 258 255 512
  \  /   \  \  \ /    \ /    |   |   \  /
   144    6  21 40    132   130 129  256
     \     \   \ |     |     |    \  /
      72    3   20     66    65   128
        \     \ /       \     \   /
         36    10       33      64
           \     \        \    /
            18     5        32
               \     \     /
                 9      16
                   \    /
                      8
                      |
                      4
                      |
                      2
                      |
                      1
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536

Crossrefs

Cf. A006577, A060413, A339991, A340008, A340418.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 0, 1 + a(
     `if`(n::even, n/2, `if`(isprime(n), 3*n+1, n-1))))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 08 2021

Mathematica

a[n_] := a[n] = If[n == 1, 0, 1 + a[
   If[EvenQ[n], n/2, If[PrimeQ[n], 3n+1, n-1]]]];
Array[a, 100] (* Jean-François Alcover, Jan 30 2021, after Alois P. Heinz *)

PARI

f(n) = if (n % 2, if (isprime(n), 3*n+1, n-1), n/2);
a(n) = my(s=n, c=0); while(s>1, s=f(s); c++); c; \\ Michel Marcus, Jan 21 2021

Python

from sympy import isprime
for n in range(1, 101):
    ct, m = 0, n
    while m > 1:
        if m%2 == 0: m //= 2
        elif isprime(m) == 1: m = 3*m + 1
        else: m -= 1
        ct += 1
    print(ct)