A359207 Number of steps to reach 0 starting with n in the map x->A359194(x) (binary complement of 3n), or -1 if 0 is never reached.

0, 1, 2, 11, 12, 1, 10, 3, 4, 13, 2, 19, 80, 9, 2, 15, 16, 81, 14, 11, 12, 1, 6, 83, 8, 73, 22, 79, 7572, 5, 18, 75, 76, 7573, 74, 7, 12, 17, 10, 3, 4, 13, 2, 7571, 4, 85, 78, 15, 96, 21, 5498, 91, 72, 13, 6, 7, 56, 13, 82, 3, 20, 5, 98, 15, 16, 21, 14, 7

Offset: 0

Author

Joshua Searle, Dec 20 2022

Keywords

• nonn
• base

Comments

It is unknown whether each positive starting integer eventually reaches 0.

From Jon E. Schoenfield, Dec 21 2022: (Start)

a(n) == n (mod 4).

a(n) = 1 iff 3*n + 1 = 4^k for some integer k. (End)

All but 10 values under 10^7 have been run to 0. Each of the remaining 10 requires over 2*10^12 steps. They're all in one group that reaches the same high value (nearly 8 million bits wide) after about 2*10^12 steps. The smallest value in this group is 3417582. - Tim Peters, Jun 14 2023

Examples

			a(7) = 3 because it takes 3 steps to reach 0: (7, 10, 1, 0).
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Tim Peters, Seeds under 2*10^6 that take more than 10^8 steps to reach 0
• Tim Peters, Seeds under 10^7 that take more than 10^9 steps to reach 0
• Joshua Searle, Collatz-inspired sequences.

Crossrefs

Cf. A035327, A359194, A359208, A359209, A359255.

Programs

Mathematica

f[n_] := FromDigits[BitXor[1, IntegerDigits[3*n, 2]], 2]; Array[-1 + Length@ NestWhileList[f, #, # != 0 &] &, 68, 0] (* Michael De Vlieger, Dec 21 2022, faster function by Hans Havermann *)

PARI

f(n) = if(n, bitneg(n, exponent(n)+1), 1); \\ A035327
a(n) = my(nb=0, m=n); while (m, m=f(3*m); nb++); nb; \\ Michel Marcus, Dec 21 2022

Python

def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
def a(n):
    i, fi = 0, n
    while fi != 0: i, fi = i+1, f(fi)
    return i
print([a(n) for n in range(68)]) # Michael S. Branicky, Dec 20 2022