# [seqfan] Binary digits change sequences

Tue Aug 13 15:27:54 CEST 2019

```Hi Everyone,

Please see the sequences below. I just want to check if theywere “legitimate” to be added to OEIS.

The definition “Number of digits that change when we multiplynatural numbers by 2 in the binary system” gives us:

2,2,2,2,4,2,2,2,4,4,4,2,4,2,2,2,4,4,4,4,6,4,4,2,4,4,4,2,4,2,2,2,4,4,4,4,6,4,4,4,6,6,6,4,6,4,4,2,4,4,4,4,6,4,4,2,4,4,4,2,4,2,2,2,4,4,4,4,6,4,4,4,6,6,6,4,6,4,4,4,6,6,6,6…….

(For example.101*10=1010. All 4 digits changed, so a(5)=4.)

I couldn’t find this sequence. However, when I divided eachterm by 2, I got A069010 (except for the zero at the start.)

Also, with the same definition but multiplying by 3instead of 2, I got

1,1,2,1,2,2,2,1,2,2,3,2,3,2,2,1,2,2,3,2,3,3,3,2,3,3,3,2,3,2,2,1,2,2,3,2,3,3,3,2,3,3,4,3,4,3,3,2,3,3,4,3,4,3,3,2,3,3,3,2,3,2,2,1,2,2,3,2,3,3,3,2,3,3,4,3,4,3,3……

Which is A007302, except for the initial zero again.

When I repeated the definition, multiplying by 4 this time, Igot

2,2,4,2,2,4,4,2,4,2,4,4,4,4,4,2,4,4,6,2,2,4,4,4,6,4,6,4,4,4,4,2,4,4,6,4,4,6,6,2,4,2,4,4,4,4,4,4,6,6,8,4,4,6,6,4,6,4,6,4,4,4,4,2,4,4,6,4,4,6,6,4,6,4,6,6,6,6,6,2….

Maybe I didn’t look well, but I couldn’t find this one evenwhen I divided by 2.
(Is multiplying by even numbers always gives us even numbers of digit changes? And if so,why?)

I am working on multiplying by more numbers. I am alsoworking on base-3,4,5,… systems.

I would appreciate any feedback.

Best,

Ali

```