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.
?!
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