Number of Ordered  

Primitive computational analyses  
Web Publication by Mountain Man Graphics, Australia  
Introduction 

The first section deals with the ability of the number of ordered factorization to account for the harmonic relationships evident in the musical Just Intonation Scale, and introduces the notion of aliquot type.
The second section examines the ancient notion of aliquot type historically (Euclid) defined by summation of the component aliquots, and notes the three separate threads of number so formalised. Abundant numbers have the sum of their divisors greater than themselves, deficient numbers have the sum of their divisors less than themselves. In the middle, with the sum of their divisors exactly equal to the number are the "Perfect Numbers".
The issue of Perfect Numbers (see above reference) and their relationship to Mersenne Primes has recently been popularised in distributed computational arrangments over the internet. For the latest in prime technology visit www.mersenne.org.
The third section introduces a new application for these ancient aliquot types. A redefinition in terms of the number of ordered factorisations (divisors) for any given number (rather than the summation of the divisors) is investigated.
That is, we define the following categories in regard to
the number of ordered factorizations.
Let the number of ordered factors of n be H(n), then:
Based on these axiomatic definitions, an analysis of the distribution of these aliquot types is undertaken for increasingly larger samples of sets of numbers, and their corresponding number of ordered factorizations.
It is clearly established by computational rather than theoretical means, that while the major percentage of the total numbers are deficient, despite the percentage of abundant numbers dimishing to an exceedingly small fraction, the major percentage of the total number of ordered factorizations is in fact contributed by this small fraction of abundant numbers.
The scale of this abundance is very surpising at first, and requires some adjustment in thinking. This avenue of thought leads to what Ray Tomes has defined as being the mainline harmonic numbers, which are essentially the greatest of these abundant numbers in any "locality" of the number environment n.
I would be interested in any theoretical feedback and/or responses to this article. Please also send any references to any other articles or notes related to ordered factorizations.
Best wishes for now,
Pete Brown
Falls Creek
Australia
Southern Autumn of 2004
The Just Intonation Scale of Music 

The elements of the Just Intonation Scale are related by their common factors, and their association with one another which clearly tends towards the maximisation of this number of ordered factorizations.
The Ancient Classification of Aliquot Type 

About 300BC, Euclid wrote at Proposition 36, Book 4, of The Elements,
concerning the aliquot parts and perfect numbers ...
About 100AD, Nichomachus elucidated this classification system for numbers:
In the following section we propose a separate but related definition
based not upon the summation,
but upon the number (ie:count)
of the ordered factorizations.
AFAIK this new definition has not before been explicitly stated.
Should this not be the case, please inform the author.
Thanks.
Reclassification of the ancient Aliquot Type 

As can be seen in the table above, the numbers 96 and 72 are "abundant".
Where N = 96, H = 112 and where N = 72, H = 76.
These Balanced numbers have not been further distinguished in the following analyses.
As an aside, Ray Tomes has conjectured (April 2004) that the largest known balanced number is the number: 2^(2^134669184)*(2^134669171). This is a number with about 10^4053946 digits, (not 4053946 digits but 10^4053946 digits!!!), but perfectly balanced, having precisely 2^(2^134669184)*(2^134669171) ordered factorizations.
Analysis of the number of Ordered Factorization by Aliquot Types 

The above analysis was then somewhat extended to all n up to 16,777,216 (or to 2^23),
and percentage calculation columns were added to discern the trends of
relative distribution within these groups of the defined aliquot types.
Analysis
to 16,777,216 by Powers of 2 





2^nn 
Prime(N) 
%P(N) 
D(N) 
%D(N) 
A(N) 
%A(N) 
Total(N) 
2^00 
0 
0.0000 
0 
0.0000 
1 
100.0000 
1 
2^01 
2 
100.0000 
0 
0.0000 
0 
0.0000 
2 
2^02 
2 
50.0000 
2 
50.0000 
0 
0.0000 
4 
2^03 
2 
25.0000 
6 
75.0000 
0 
0.0000 
8 
2^04 
5 
31.2500 
11 
68.7500 
0 
0.0000 
16 
2^05 
7 
21.8750 
24 
75.0000 
1 
3.1250 
32 
2^06 
13 
20.3125 
48 
75.0000 
3 
4.6875 
64 
2^07 
23 
17.9688 
101 
78.9063 
4 
3.1250 
128 
2^08 
43 
16.7969 
206 
80.4688 
7 
2.7344 
256 
2^09 
75 
14.6484 
426 
83.2031 
11 
2.1484 
512 
2^10 
137 
13.3789 
868 
84.7656 
19 
1.8555 
1,024 
2^11 
255 
12.4512 
1765 
86.1816 
28 
1.3672 
2,048 
2^12 
464 
11.3281 
3585 
87.5244 
47 
1.1475 
4,096 
2^13 
872 
10.6445 
7249 
88.4888 
71 
0.8667 
8,192 
2^14 
1,612 
9.8389 
14659 
89.4714 
113 
0.6897 
16,384 
2^15 
3,030 
9.2468 
29562 
90.2161 
176 
0.5371 
32,768 
2^16 
5,709 
8.7112 
59568 
90.8936 
259 
0.3952 
65,536 
2^17 
10,749 
8.2008 
119904 
91.4795 
419 
0.3197 
131,072 
2^18 
20,390 
7.7782 
241125 
91.9819 
629 
0.2399 
262,144 
2^19 
38,635 
7.3690 
484674 
92.4442 
979 
0.1867 
524,288 
2^20 
73,586 
7.0177 
973533 
92.8433 
1457 
0.1390 
1,048,576 
2^21 
140,336 
6.6917 
1954482 
93.1970 
2334 
0.1113 
2,097,152 
2^22 
268,216 
6.3948 
3922629 
93.5228 
3459 
0.0825 
4,194,304 
2^23 
513,708 
6.1239 
7869549 
93.8123 
5352 
0.0638 
8,388,609 








Totals 
1,077,871 
6.4246 
15683976 
93.4838 
15369 
0.0916 
16,777,216 
In summary we might note that the deficient numbers (along with a dimishing number of prime numbers) dominate the first 16.777 million integers, with 99.9% of number. The percentage of the abundant ordered factorizations in this range can clearly be seen to represent only the modest percentage of 0.0916.
Analysis
to 16,777,216 by Powers of 2 





2^nn 
Tot(H) 
%D(H) 
Deficient(H) 
%A(H) 
Abundant(H) 
%P(H) 
Prime(H)=P(N) 
2^00 
1 
0.0000 
0 
100.0000 
1 
0.0000 
0 
2^01 
2 
0.0000 
0 
0.0000 
0 
100.0000 
2 
2^02 
7 
71.4286 
5 
0.0000 
0 
28.5714 
2 
2^03 
25 
92.0000 
23 
0.0000 
0 
8.0000 
2 
2^04 
85 
94.1176 
80 
0.0000 
0 
5.8824 
5 
2^05 
289 
80.9689 
234 
16.6090 
48 
2.4221 
7 
2^06 
969 
65.6347 
636 
33.0237 
320 
1.3416 
13 
2^07 
3,237 
65.8017 
2,130 
33.4878 
1,084 
0.7105 
23 
2^08 
10,759 
58.3326 
6,276 
41.2678 
4,440 
0.3997 
43 
2^09 
35,670 
57.2778 
20,431 
42.5119 
15,164 
0.2103 
75 
2^10 
118,167 
50.5099 
59,686 
49.3742 
58,344 
0.1159 
137 
2^11 
391,526 
48.5033 
189,903 
51.4316 
201,368 
0.0651 
255 
2^12 
1,297,529 
43.9689 
570,509 
55.9954 
726,556 
0.0358 
464 
2^13 
4,300,882 
40.4781 
1,740,914 
59.5017 
2,559,096 
0.0203 
872 
2^14 
14,255,826 
37.3621 
5,326,270 
62.6266 
8,927,944 
0.0113 
1,612 
2^15 
47,244,938 
33.7473 
15,943,894 
66.2463 
31,298,014 
0.0064 
3,030 
2^16 
156,539,656 
32.2364 
50,462,683 
67.7600 
106,071,264 
0.0036 
5,709 
2^17 
518,573,412 
28.8093 
149,397,519 
71.1886 
369,165,144 
0.0021 
10,749 
2^18 
1,717,669,071 
26.9077 
462,185,473 
73.0911 
1,255,463,208 
0.0012 
20,390 
2^19 
5,689,131,636 
24.3757 
1,386,764,569 
75.6236 
4,302,328,432 
0.0007 
38,635 
2^20 
18,843,448,795 
22.8560 
4,306,860,069 
77.1436 
14,536,515,140 
0.0004 
73,586 
2^21 
62,417,265,668 
20.3462 
12,699,535,156 
79.6536 
49,717,590,176 
0.0002 
140,336 
2^22 
206,771,806,824 
18.8015 
38,876,235,196 
81.1984 
167,895,303,412 
0.0001 
268,216 
2^23 
685,060,801,921 
17.0794 
117,004,573,089 
82.9205 
568,055,715,124 
0.0001 
513,708 








Totals 
981,242,896,895 
17.8304 
174,959,874,745 
82.1695 
806,281,944,279 
0.0001 
1,077,871 
In summary we might note that in the first 16.777 million integers
the abundant ordered factorizations are represented by only the modest percentage of 0.0916% (of numbers),
yet they account for 82.1695% of the total number of ordered factorizations.
Summary and Conclusions 

These results suggest that the ancient concept of "abundance" may be validly applied to more than the summation of factors. The new definition  by number of ordered factorizations  of deficient and abundant appears to be vindicated by the above analysis, and suggests further research might prove rewarding.
These results also support the hypotheses outlined in the Theory of Harmonics by Ray Tomes, in which critical emphasis is placed upon a defined harmonic mainline series which is represented by the most abundant of the abundant ordered factorizations.
PRF Brown
Falls Creek
Australia