# [seqfan] Re: Binary digits change sequences

Tue Aug 13 21:44:32 CEST 2019

```Ali,

1) Your sequences should also have initial zeros. Multiplying zero by anything gives zero, which has no digits changing, so a(0) = 0.

2) The values will always be even when you are multiplying by a power of two. Otherwise, not.

3) If a(n) is the sequence multiplying by 2, and b(n) is the sequence multiplying by 4, then:

b(n) = a(A059905(n)) + a(A059906(n)).

Take some time to understand where this formula is coming from.

4) Formulas similar to (3), but with more terms, will apply for other powers of whatever base you are working in.

5) For multipliers that are not a power of the base, I would expect no such simple relationship.

6) Certainly these are appropriate for the database. Don't overdo it; 6 to 10 examples would be (IMO) sufficient.

-----Original Message-----
From: Ali Sada via SeqFan <seqfan at list.seqfan.eu>
To: Sequence Fanatics Discussion List <seqfan at list.seqfan.eu>
Cc: Ali Sada <pemd70 at yahoo.com>
Sent: Tue, Aug 13, 2019 1:24 pm
Subject: [seqfan] Binary digits change sequences

Hi Everyone,

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

The definition “Number of digits that change when we multiply
natural 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 each
term 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

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