# [seqfan] Ulam-related request

Chris Starling chaosorder4 at gmail.com
Sat May 30 03:44:29 CEST 2009

```Greetings, Seqfans, Romans, Countrymen:

My name is Chris, and I'm a hobbyist in math and art, and at finding their
junctions.  However, I'm still 96-98% ignorant in both areas.  I could use

A goal of mine is to create a beautiful  Ulam-inspired graph showing a good
chunk of integers' inherent or "causative" fingerprints through
divisibility.  More specifically, I want to build an Ulam winding, but with
composites, and up into 3D, so that each integer gets an urban spire
proportional to its factorization totals and colored by breakdown.

As for the input data for this, I have a good start in the sequences...

http://www.research.att.com/~njas/sequences/A001055<http://www.research.att.com/%7Enjas/sequences/A001055>

Number of ways of factoring n (for n>1 we require that all the factors are
greater than 1).
(Formerly M0095 N0032)
and

http://www.research.att.com/~njas/sequences/A050322
<http://www.research.att.com/%7Enjas/sequences/A050322>

Number of factorizations indexed by prime signatures:
A001055<http://www.research.att.com/%7Enjas/sequences/A001055>
(A025487 <http://www.research.att.com/%7Enjas/sequences/A025487>).

which show the totals, but I can't bear the thought of building this without
breakdown by parts quantity.  I can't seem to find a table of this or
reference to its definite presence in a book.  I'd be quite happy to
calculate them myself from prime signatures but alas I lack the terminology
and understanding for a paper on it, containing a formula, that I found and

(Counting number of factorizations of a natural number
Shamik Ghosh
Department of Mathematics, Jadavpur University, Kolkata - 700 032, India.)

Could someone(s) please point me in the right direction(s)?

My initial goal for this is to get out to at least 1027, the first hexagon
over 1000.  I plan a virtual sculpture before a tangible.  Each integer's
factorizations spire would have segments proportional to each parts
quantity.

Incase they may be of interest, here are pics of composites along with
primes in Ulam windings, represented merely by their *pairs* quantities.
There is a square out to 10,000 and a hexagon and triangle out to just past
1,000:

http://or9an1c.deviantart.com/gallery/#Mathema-Tikal

I should mention, I cannot be absolutely certain of each integer's
representative value being correct.  The images are intended for a general
impression, or for inspiration.

Also, here are my contributions to the OEIS, which involve figurate
polyhedra, truncated into new forms, sometimes requiring a 2-variate table:

http://www.research.att.com/~njas/sequences/?q=chaosorder4&sort=0&fmt=0&language=english&go=Search<http://www.research.att.com/%7Enjas/sequences/?q=chaosorder4&sort=0&fmt=0&language=english&go=Search>

BTW - does anyone know of an Ulam winding where the winding itself is in 3D,
forming a figurate polyhedron?  Someone(s) must have thought of that before,
but I cannot find any such construction or even mention.