A261690 a(1) = 1; for n>1, a(n) is the smallest number not already present which is entailed by the rules (i) k present => 3*k+1 present; (ii) 2*k present => k present.

1, 4, 2, 7, 13, 22, 11, 34, 17, 40, 20, 10, 5, 16, 8, 25, 31, 49, 52, 26, 61, 67, 76, 38, 19, 58, 29, 79, 88, 44, 94, 47, 103, 115, 121, 133, 142, 71, 148, 74, 37, 112, 56, 28, 14, 43, 85, 130, 65, 157, 169, 175, 184, 92, 46, 23, 70, 35, 106, 53, 139, 160, 80

Offset: 1

Author

Vladimir Shevelev, Aug 28 2015

Keywords

• nonn

Comments

An analog of A109732 such that the statement 'the sequence is a permutation of the positive integers not divisible by 3' is equivalent to the (3*n+1)-conjecture for numbers not divisible by 3.

On Aug 29 2015, Max Alekseyev noted that, while the (3*n+1)-conjecture indeed implies that the sequence is a permutation of the positive integers not divisible by 3, the opposite statement is an open question. The author cannot yet prove this, so his previous comment is only a conjecture.

In connection with this, consider the following conjecture which could be called the (n-1)/3-conjecture. Let n be any number not divisible by 3. If n==1 (mod 3) and (n-1)/3 is not divisible by 3, then set n_1 = (n-1)/3. Otherwise set n_1 = 2*n. Conjecture. There exists an iteration n_m = 1. Does the (n-1)/3-conjecture imply the (3*n+1)-conjecture?

Example: 19->38->76->25->8->16->5->10->20->40->13->4->1.

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006577, A006877, A006884, A109732.