# [seqfan] Complementary series to A226383 - Collatz related

Joe Slater seqfan at slatermold.com
Mon Dec 19 09:23:29 CET 2016

```Series A226383 in the OEIS consists of those numbers that have at least one
3-smooth representation that is special of level k. That is, only those
numbers that can be expressed as the sum of (3^0 * 2^a0) + (3^1 * 2^a1) ...
(3^k *2^ak); with a0 > a1 > ... ak, and ak = 0. Every number in this series
is the sum of at least one power of two and at least one power of three, so
no numbers in the series are either multiples of two or of three. The
series is interesting because (if the Collatz Conjecture is correct) every
odd number can be represented as (2^j - A226383(n))/3^k, as long as we
include the number 1 as part of the series A226383.

The first 25 elements of A226383-including-1 are:
1, 5, 7, 11, 19, 19, 23, 29, 31, 35, 37, 47, 49, 53, 65, 65, 67, 73, 79,
85, 85, 89, 89, 97, 101.

Consider the complementary series, omitting multiples of two and multiples
of three: 13, 17, 25, 41, 43, 55, 59, 61, 71, 77, 83, 91, 95 ...

There are hints of a pattern in this: the first four numbers are separated
by 4, 8, 16; but the next number is 73, which is a term in A226383. Some
(but not all!) multiples of this series appear in A226383. Does anyone know
how to calculate these numbers directly? Is it an infinite series?
Are there other interesting things we can say about it?

Joe Slater

```