Riffs & Rotes & Numbers Round & Rare & Random

[Jon Awbrey]

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Roundness & Round Numbers

I find these entries in the database:

%H A001222 E. W. Weisstein, Link to a section of The World of Mathematics.
%N A048597 Very round numbers: reduced residue system consists of only primes and 1.

But, of course, the link to WOM is currently missing.

Here is the tidbit from Hardy & Wright that I mentioned:

| 22.12  A note on round numbers.  A number is usually called "round"
| if it is the product of a considerable number of comparatively small
| factors.  Thus 1200 = 2^4 . 3 . 5^2 would certainly be called round.
| The roundness of a number like 2187 = 3^7 is obscured by the decimal
| notation.
|
| It is a matter of common observation that 'round numbers are very rare';
| the fact may be verified by any one who will make a habit of factorizing
| numbers which, like numbers of taxi-cabs or railway carriages, are presented
| to his attention in a random manner.  Theorem 431 contains the mathematical
| explanation of this phenomenon.
|
| Either of of the functions omega(n) or Omega(n) gives a natural measure of
| the 'roundness' of n, and each of them is usually about loglog n, a function
| of n which increases very slowly.  Thus loglog 10^7 is a little less than 3,
| and loglog 10^80 is a little larger than 5.  A number near 10^7 (the limit of
| the factor tables [in 1938? 1979?]) will usually have about 3 prime factors;
| and a number near 10^80 (the number, approximately, of protons in the universe
| [in 1938? 1979?]) about 5 or 6.  A number like 6092087 = 37 . 229 . 719 is in
| a sense a "typical" number.
|
| These facts seem at first very surprising, but the real paradox lies
| a little deeper.  What is really surprising is that most numbers should
| have 'so many' factors and not that they should have so few.  Theorem 431
| [which says that the normal order of omega(n) and Omega(n) is loglog n]
| contains two assertions, that omega(n) is usually not much larger than
| loglog n and that it is usually not much smaller;  and it is the second
| assertion which lies deeper and is more difficult to prove.  That omega(n)
| is usually not much larger than loglog n can be deduced from Theorem 430
| without the aid of (22.11.6).
|
| Hardy, G.H. & Wright, E.M.,
|'An Introduction to the Theory of Numbers',
| Fifth Edition, Oxford University/Clarendon Press,
| Oxford, UK, 1979.  (First Edition 1938).
| pages 358-359.

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> 
> I A062537
> %S A062537 0,1,2,2,3,3,3,3,3,4,4,4,4,4,5,3,4,4,4,5,5,5,4,5,4,5,4,5,5,6,5,4,6,5,6,
> %T A062537 5,5,5,6,6,5,6,5,6,6,5,6,5,4,5,6,6,4,5,7,6,6,6,5,7,5,6,6,4,7,7,5,6,6,7,
> %U A062537 6,6,6,6,6,6,7,7,6,6,4,6,5,7,7,6,7,7,6,7,7,6,7,7,7,6,5,5,7,6,6,7,5,7,8
> %N A062537 Nodes in riff (rooted index-functional forest) for n.
> %C A062537 A061396(n) gives number of times n appears in this sequence.
> %H A062537 J. Awbrey, <a href="http://www.research.att.com/~njas/sequences/a061396a.txt">
>            Illustrations of riffs for small integers</a>
> %F A062537 a(PROD(p_i^e_i)) = SUM(a(i)+a(e_i)+1), product over nonzero e_i in prime factorization.
> %K A062537 nonn,easy,nice,new
> %O A062537 1,3
> %A A062537 dww, Jun 25, 2001

Please excuse the pre-caffeinated mistakes below.
Since I can never seem to remember when to regard
0 as a natural number, I am almost tempted to start
using M for the "multiplicative naturals" 1, 2, 3, ...
and N for the "additive naturals" 0, 1, 2, 3, ...
 
> Let N = positive integers.  [no, nonn].
> A062537 : N -> N such that
> A062537 : n ~> |riff(n)| = Cardinality (Points(riff(n))),
> AKA the "weight", or the number of nodes in the riff of n.
> This is what I was referring to as a "measure of complexity" on the positive integers.
> Some of the only work that I was able to find on simlilar notions was in Hardy & Wright,
> where there was a little bit on what they called measures of "roundness".  Of course,
> if we want to instantly double the number of sequences in the EIS, we could simply
> now form the functional composition A062537(A-whatever) = |riff(A-whatever)|.
> Exercise for the reader.  But of course, again, some of the more interesting
> cases would be where it does not quite double, but coincides with sequences
> that are already indexed.  The mind boggles.
> 
> The mindset of the graph counting circle where I was doing this work was so focussed
> on the enumeration functions themselves, nice monotone sequences and their asymptotics,
> that I would not even have regarded this as a sequence, but now that it is, it encourages
> me to go back and look at those other measures of complexity, based on more familiar families
> of graphs, like the rooted trees in Göbel's correspondence, or the planted plane tree measures
> that I was looking at.  So I will go see what I can dig up from the runes.
> 
> Jon Awbrey
> 
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