A386314 a(1) = 1 and thereafter a(n) is the smallest number k of the form 6*x+-1 not already in the sequence but where the reduced Collatz step A139391(k) is in the sequence.

1, 5, 13, 17, 11, 7, 29, 19, 25, 37, 49, 53, 35, 23, 61, 65, 43, 77, 85, 101, 67, 89, 59, 113, 133, 149, 157, 173, 115, 181, 197, 131, 205, 209, 139, 185, 229, 241, 245, 163, 217, 269, 179, 119, 79, 277, 289, 301, 305, 203, 317, 211, 281, 187, 325, 341, 227, 151, 349, 373

Offset: 1

Author

Jules Beauchamp, Jul 18 2025

Keywords

• nonn
• new

Comments

These numbers are the Collatz pre-images in the form 6*x +- 1 of all previous terms not already in the sequence.

The pre-images of a term t are all p which reach t by a single odd to odd step A139391(p) = t.

These pre-images are those p = (t*2^k-1)/3 with k>=0 which are odd integers, and with here t != 0 (mod 3) there are infinitely many p != 0 (mod 3) for each t.

Multiples of 3 have no odd pre-images and are excluded here in order to have the essential part of the tree of odd to odd descents.

The trajectory of a term t reaches 1 by steps to successively earlier terms in this sequence (at various distances apart).

If the Collatz conjecture is true, then this sequence is permutation of the numbers of the form 6x +- 1 (A007310).

Examples

			a(3) = 13, since 13 (a pre-image of a(2) = 5) is the smallest unused pre-image of a(1) and a(2).
a(10) = 37 since 37 (a pre-image of a(6) = 7) is the smallest unused pre-image of all previous terms.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (using Michel Marcus' code)
• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A007310, A178414, A178415, A139391.

Programs

PARI

lista(nn) = my(va=List(1), vs = Map(), imin=1, i=imin, nb=1); mapput(vs, 1, 1); while(#vaMichel Marcus, Aug 25 2025