[seqfan] Re: Balanced numbers

[M. F. Hasler]

On Sat, Mar 14, 2015 at 6:14 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
> Hello SeqFans,
> We look here for "balanced" integers
> that are _not_ palindromic.
>
> S =
> 1030,1140,1250,1360,1412,1470,1412,
>
> 1933 is "balanced" because the two halves
> of 1933 have the same weight:
>
> 19=33 when (1x3/2)+(9x1/2)=(3x1/2)+(3x3/2)

Proposed as https://oeis.org/draft/A256075

> There is of course an infinite number
> of "balanced" integers (as 1933, for
> instance, can be extended to 119331, or 619336, etc.)

One could call "primitive" balanced numbers which are not of the form
a***a.

> If S is an interesting seq, why not
> compute T, the "balanced primes" seq?

This is proposed as https://oeis.org/draft/A256076

The same can be done for other bases, e.g. in base 2:

70,78,150,266,282,294,310,334,350,355,371,397,413,540,554,582,630,686,723,798,813,

70 = 1000110[2] is balanced because
1*3 = 1*1 + 1*2

We see that only the 10-th non-palindromic balanced number is odd, and
the same is true in base 2 (but neither in base 3
(87,96,105,137,146,155,169,178,...) nor in base 4
(76,114,141,179,206,264,280,296,312,...).

> [BTW is there a 10-digit balanced
> integer with no repeated digit?]

It seems that there is no such number !

The least 1-9 pandigital balanced number seems to be 137986542.

Best,
Maximilian