A direct formula for counting 1s in binary form of a positive integer

Asked Jan 30 '21 at 16:37

Active Jan 30 '21 at 16:56

Viewed 53 times

The question is edited based on the @Damien comment.

Question:

Is there a direct formula (not recursive) for counting 1s in binary form of a positive integer?

For example, suppose the formula is called $f(n)$ for a positive integer $n$. Then, we have

$f(3)=2$, $f(13)=3$ and $f(16)=1$, since we have $[3]_2=11$,$[13]_2=1101$ and $[16]_2=10000$.

Thank you in advance

edited Jan 30 '21 at 16:56

asked Jan 30 '21 at 16:37

user0410

2

This is A000120 in OEIS. – lulu Jan 30 '21 at 16:39
@lulu It is a sequence and not a formula. Am I right? – user0410 Jan 30 '21 at 16:43
Which type of formula do you want? For example $f(n) = f(n/2) + n \mod 2$ and $f(0)=0$ – Damien Jan 30 '21 at 16:45
1

From the discussion and references attached to that OEIS link, you can get a sense of what is known about computing it. No simple closed formula, for instance, but it does have nice "self replicating" properties which are discussed. – lulu Jan 30 '21 at 16:49
@Damien A direct formula. The $f(n) = f(n/2) + n \mod 2$ is a recursive one. Thunks – user0410 Jan 30 '21 at 16:49
@user0410 You should put this kind of information in the post – Damien Jan 30 '21 at 16:51
@lulu You are right. Thanks for the proposed link. – user0410 Jan 30 '21 at 16:52
@Damien Based on your suggestion, the question will be edited. Thanks – user0410 Jan 30 '21 at 16:53
This question has already been asked: see here. – Fabius Wiesner Jan 30 '21 at 20:26

0 Answers0