# [seqfan] Re: Length of binary representation of n^3

M. F. Hasler oeis at hasler.fr
Tue Oct 27 20:22:18 CET 2020

```On Tue, 27 Oct 2020, 14:16 JEREMY GARDINER via wrote:

> A070939 Length of binary representation of n.
> A004221 10*log_10 (n) rounded up.
>
> Calculating A070939(n^3) I noticed some similarity to the terms of
> A004221(n).
>

This is because
log_2 ( n^3 ) = 3 log_10 ( n ) / log_10 ( 2 ) = 9.966 log_10 ( n )
because as we know well,
log_10 ( 2 ) = 0.30103
is very close 3/10.
So when this and 10 log_10 ( n ) are rounded up, it gives often the same as
long as the result is not larger than ~ 10^2, but not beyond that.

- Maximilian

For the values of n where A070939(n^3) differs from A004221(n) I find:
>
>
> 1,20,40,80,101,126,127,159,160,161,200,201,202,203,252,253,254,255,317,318,319,320,321,322,399,400,401,402,403,404,405,406,502,503,504,505,506,507,508,509,510,511,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,
>
> I'm happy to submit this sequence if anyone thinks it is interesting
> enough.
>
> Regards,
>
> Jeremy Gardiner
>

```