# yet another unexpected EIS hit

Marc LeBrun mlb at well.com
Thu Dec 13 08:22:02 CET 2001

```Here's a pretty pair of dots that someone might be able to connect:

Define a "numbral arithmetic" by replacing addition with binary bitwise
inclusive-OR (so that [3] + [5] = [7] etc) and multiplication becomes
shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc).

These numbrals have some interesting interpretations (such as a kind of
"infinite base" system, or alternatively as sets of integers) that for
brevity I'll resist  presenting.

Anyway, we can say naturally that [d] divides [n] when there exists an [e]
such that

[d] * [e] = [n].

For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14].

Dot X: Counting the proper divisors of [2^n-1] gives the sequence

0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, ...

Dot Y: But this appears to be A048888, the anti-diagonal sums of table
A048887 (qv)

1  1  1  1  1  1  1 ...
1  2  3  5  8 13 ...
1  2  4  7 13 ...
1  2  4  8 ...
...

where A(i,j) is the number of compositions of j into parts all <=i.

?!

```