# [seqfan] Balanced numbers

Eric Angelini Eric.Angelini at kntv.be
Sat Mar 14 11:14:27 CET 2015

```Hello SeqFans,
We look here for "balanced" integers
that are _not_ palindromic.

S =
1030,1140,1250,1360,1412,1470,1412,
1522,1580,1603,1622,1690,1713,1742,
1823,1852,1904,1933,1962,2070,...

1933 is "balanced" because the two halves
of 1933 have the same weight:

19=33 when (1x3/2)+(9x1/2)=(3x1/2)+(3x3/2)

We consider here that a "3" far from
the "center" of the integer has more
weight than a "3" close to the center.
The weight-multiplier is simply the said
distance (to the center).

So, an interger with digits "abcd" is
evaluated like this:

...5/2   3/2   1/2   1/2   3/2   5/2...
a       b      c     d

(ax3/2)+(bx1/2) = (cx1/2)+(dx3/2)

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

The "distance to the center" was
fixed here to (2k+1) multiples of 1/2
because the above integers have an
even quantity of digits.

Should the integer have an odd quantity
of digits, then would the distance be
fixed to k multiples of 1:

10020 is balanced, as are 10920 or
7468429.

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

Best,
É.
[BTW is there a 10-digit balanced
integer with no repeated digit?]

```