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

Author

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

Keywords

• nonn

Comments

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A075677, A075680, A006667.

All terms are in A016945.

Programs

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,2, t=oddsteps(k); if(t<=n && v[t]==0, v[t]=k; if(r-- == 0, return(v)))) \\ Charles R Greathouse IV, Dec 22 2021

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.