# [seqfan] Re: binary expansion 1+0*

franktaw at netscape.net franktaw at netscape.net
Mon Mar 22 23:06:03 CET 2010

```But there is another property: these are the numbers such that
mod(2^k,n) is a power of 2 for any non-negative k. Thus, by my "rule of
2" (if there are 2 marginally interesting, not-obviously-related facts
about a sequence, it is worth including) it should be in the OEIS.  In
this case, the relationship is a little closer than optimal, but the
second property is a bit more than marginally interesting.

Note: Joerg stated only that these numbers have the modular property,
not that they are the only numbers that do. But, in fact, they are.

Note first the definition of A023758: differences of two distinct
powers of 2. Any member of A023758 that is not a power of 2 can be
written in the form 2^k-2^j, where k > j+1. If n is not a power of 2,
and 2^k is the smallest power of 2 greater than n, then mod(2^k,n) =
2^k-n, and if this is power of 2 -- 2^j, n = 2^k-2^j. And if n is a
power of 2, mod(2^k,n) = 0 for any 2^k >= n.

So, I would say yes, this should be submitted, being sure to include
the modular property and the cross-reference to A023758.

-----Original Message-----
From: Graeme McRae <g_m at mcraefamily.com>

Is there any need for a new sequence that could be defined as
{A023758} - {A000079} ?

--Graeme McRae,
Palmdale, CA,
Sent from my iPhone

On Mar 22, 2010, at 11:51 AM, Joerg Arndt <arndt at jjj.de> wrote:

> I erred, I wanted to exclude the powers of 2.
> The pattern is 11+0* (and the initial 1 has
> to be omitted in my seq).
>
> Thanks for correcting!

```