[seqfan] n-binary-hypercubes and A138553.

Eric moongerms at wanadoo.fr
Sun Apr 29 19:16:40 CEST 2012


There are some years, Franklin T. Adams-Watters has been able to form an
idea of sequence.

This is A138553 Table, T(n,k) is the number of divisors of n that are <= k.

Row lengths for this table are A003418 :

a(n) = least common multiple (or lcm) of {1, 2, ..., n}
[1, 1, 2, 6, 12, 60, 60, 420, 840, ...]

Using the same principle of game with 1 and 0,
applied to the coordinates of a n-binary-hypercube,
it appears, i found in the OEIS, that for row lengths 2^n A000079, sums of
0's and 1's gives the sequence A048881.

Now I ask myself whether the binary combinations from A138553 are
path/distance in n-binary-hypercubes,
and what coordinates are used or not in these paths in other dimensions.

For exemple,

in 2D All points are "used".
x 0101
y 0011


in 3D (0,0,1) and (0,1,1) are "not used"
the frequency for (1,0,0) (0,1,0) (1,0,1) is 2 (the path passes twice
through these points)
the frequency for (1,1,0) (1,1,1) is 1

x 01010101
y 00110011
z 00001111


et cetera...

I guess a lot depends on the formalism by starting the counting.
However I was wondering just the question of whether the amount of unused
coordinates would grow indefinitely. Or other properties.

Maybe this could "inspire" people more competent than me, and why not
comment or create sequence. (why not too fire me from the list ^^)

Anyway thank you to those who provide day after day maintenance of the
database of the OEIS.
I love to follow all the OEIS cards.

Best Regards.

(we avoided the worst. :o) My first idea was much more twisted. I imagined a
stack of cylinders of different perimeters equal to A003418, each cylinder
consists of several rotative rings made with opaque and transparent boxes,
and a few beams of light here and there, or something like that.)

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