Mersenne numbers and the (divisors) property.

[Annette.Warlich at t-online.de]

Am 28.04.05 05:34 schrieb alexandre.wajnberg at skynet.be:

> 
> Another thing to note (I don't know if it's of interest for your
> discussion) is that A007733 is a fractal sequence, related to A002326:
> 
> A007733:
> 1,1,2,1,4,2,3,1,6,4,10,2,12,3,4,1,8,6,18,4,6,10,11,2,20,12,18,3,28,4,5,1,10,8,12,6,36,18,12,4,20,6,14,10,12,11,23,2,21,20,8,12,52,18,20,3,18,28,58,4,60,5,6,1,12,10,66,8,22,12,35,
> 6,9,36,20,18,30,12,39,4,54,20,82,6
> 
> The even-index terms of this sequence are this sequence itself,
> constructed on A002326, whose terms are the odd-index terms of this
> sequence.
> 
> Alexandre

Well, you could write it in this table (which can be extended
in the obvious way)

    N              |       A(N)  (notation from right to left)
-----------------------------------------------
               9   |   6
        5     10   |   4   4
              11   |  10
    3   6     12   |   2   2  2
              13   |  12
        7     14   |   3   3
              15   |   4
2   4   8     16   |   1   1  1   1

You can see the progression in A(N) for N= 2^k-1   which is
A(2^k-1) = k and

    N              |       A(N)  (notation from right to left)
-----------------------------------------------
               9   |
        5     10   |
              11   |
    3   6     12   |          2
              13   |
        7     14   |       3
              15   |   4
2   4   8     16   |


A(2^k+1) = 2k

    N              |       A(N)  (notation from right to left)
-----------------------------------------------
               9   |   6
        5     10   |       4
              11   |
    3   6     12   |          2
              13   |
        7     14   |
              15   |
2   4   8     16   |


and also expect the "interleaving" of the sequence by itself resp.
indexes of powers of 2, since the lengthes of N and 2^k*N are all
the same.

Gottfried Helms