A349873 Smallest odd value such that any Collatz trajectory in which it occurs contains exactly n odd values other than '1'.

21, 3, 69, 45, 15, 9, 51, 33, 87, 57, 39, 105, 135, 363, 123, 339, 219, 159, 393, 519, 681, 897, 603, 111, 297, 1581, 1053, 351, 933, 621, 207, 549, 183, 243, 645, 429, 285, 189, 63, 165, 27, 147, 195, 129, 171, 231, 609, 411, 543, 1449, 975, 327, 873, 1185, 1527, 1017

Offset

1,1

Comments

a(n) necessarily is the first odd term in any Collatz trajectory in which it occurs.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

a(n) mod 6 = 3 for all n>0. The odd multiples of 3 form the 'Garden-of-Eden' set (terms without a predecessor) under iterations of the reduced Collatz function A075677.

Example

a(1)=21 as 21 occurs solely in Collatz trajectories starting with 21*2^k, and these trajectories all contain one single odd value other than 1. No value smaller than 21 satisfies these requirements. In particular, a(1) does not equal 5 since 5 is part of Collatz trajectories that contain multiple odd values other than 1 (e.g., ...,13,40,20,10,5,16,8,4,2,1).
a(2)=3 as 3 occurs solely in Collatz trajectories starting with 3*2^k, and these trajectories all contain exactly two odd values other than 1 (namely 3 and 5).

Prog

(PARI)
oddsteps(n)={my(s=0); while(n!=1, if(n%2, n=(3*n+1); s++); n/=2); s}
a(n)={forstep(k=3, oo, 6, if(oddsteps(k)==n, return(k)))} \\ Andrew Howroyd, Dec 19 2021
(PARI) oddsteps(n)=my(s); while(n>1, n+=n>>1+1; if(!bitand(n, 1), n >>= valuation(n, 2)); s++); s
first(n)=my(v=vector(n), r=n, t); forstep(k=3, oo, 6, t=oddsteps(k); if(t<=n && v[t]==0, v[t]=k; if(r-- == 0, return(v)))) \\ Charles R Greathouse IV, Dec 22 2021

Crossrefs

Cf. A075677, A075680, A006667, A092893.

All terms are in A016945.

Sequence in context: A118389 A077690 A018855 * A372282 A263277 A040428

Adjacent sequences: A349870 A349871 A349872 * A349874 A349875 A349876

Keyword

nonn

Author

Johannes M.V.A. Koelman, Dec 03 2021

Status

approved