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Prime Signature

Page history last edited by Bill McEachen 6 years, 9 months ago

this is my deleted edit from Wikipedia illustrating a practical example for Prime Signatures (http://en.wikipedia.org/wiki/Prime_signature)

 

An example of the use of prime signatures is as follows.  The entries of (http://oeis.org/A129912 ) factor as:

 

entry  xpansion  A=# prime factors  B=Exp sum  Product A*B
2*3 
12  2^2*3 
30  2*3*5 
60  2^2*3*5  12 
180  2^2*3^2*5  15 
210  2*3*5*7  16 
360 2^3*3^2*5 3 6 18
420 2^2*3*5*7 4 5 20
1260 2^2*3^2*5*7  4 6 24
2310  2*3*5*7*11 5 5 25
2520 2^3*3^2*5*7 4 7 28
4620  2^2*3*5*7*11 5 6 30
6300 2^2*3^2*5^2*7 4 7 28
12600 2^3*3^2*5^2*7 4 8 32
13860  2^2*3^2*5*7*11 5 7 35

 

The embedded signature string (last column) can then be searched for in OEIS, and the independent (http://oeis.org/A071562/list ) is found to highly correlate:

 

                    1,2,4,6,8,9,12,15,16,18,20,24,25,28,30,32,35,...

 

[ I have had someone criticize the lack of early term match up, but of course this is often seen in many cases]

 

****************

To preserve my comments from Wikipedia, here are a section:

The "embedded string" referred to is just the number sequence of the last column, effectively ordered.

The prime factor expansion of course is directly related to the prime signature, in particular for primorial sequences which have their exponents ordered in a specific way. The sequences used are not particularly relevant other than they are 2 for which such a connection existed. As signatures had been clearly explained just prior, it seems redundant to list them in the table.

For the example used, the practical use was in confirming the precise entry preceding the entry=100th primorial (selected as its "product" is square ie 100*100=10000) Via the signature linkage, one knows that connected A071532 terms are ...9991,9996,9999,10000,...

Someone had suggested that the preceding term could be 4705968353749845378343279973064195863536763443315426361801935164371167456947236 332613861927611714883983648469710968282522106130698506603418487862612958364179972073224374325956239485314766284153514139041745896918076382000.

Using the signature mechanism, seeing that the candidate is the product of the 1st, 5th, 34th and 78th primorials, the resulting product is only (1+5+34+78)*(78)=9204<<9999, from which one can deduce it (and others) are incorrect without having to specifically compute the candidates rigorously.--Billymac00 (talk) 03:53, 10 January 2008 (UTC)

 

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