Study Of The Lcm And Shcn Series

Study Of The Lcm And Shcn Series

OctarineBean
OctarineBean

Nov 3 2016, 03:21 AM #1

The idea occurred to me to study the "mysterious", praised, highly composite numbers by comparing the exponents to other divisible series of numbers.

At one end is the powers of 2 (1,2,4,8,16,32...) and the opposite is the primorials (1,2,6,30,210,2310...) and in-between I put some other series sorted by how much of a "long tail" each had. First I put in the factorial series (1,2,6,24,120,720...) which has the first seven as HCN and five of these as SHCN, but after that does not seem to have any again; they pick up 2's and small primes too quickly to fit the distribution of HCN...

Then I gave a good look at the series of least common multiples of 1 through N. When you figure them by hand you could find the first ten in a few minutes: 1**, 2**, 6**, 12**, 60**, 420, 840*, 2520**, 27720*, 360360*, 720720**... an asterisk indicates a HCN and two indicates a SHCN. So the first members of this series are dense with divisors with some gaps, but how does the pattern develop? The easy to figure out formulation for a LCM number is to multiply every highest power of each prime that is underneath some constant:

LCM(n) = product across all prime p of p^floor(n / log p)

while the HCNs are more unpredictable, they do not occur strictly as multiples of each other and so they do not readily give a formula to produce all of them. However, some websites told me to try:

f(n) = product across all prime p of p^floor(1 / (p^(1/n) - 1) )

and when I used this with a real input less than four I got these thirteen numbers from it:

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600

which are exactly the first thirteen SHCNs. The function inside the exponent increases everywhere along with n, and adds new primes one at a time according to a distribution. I believe that Erdos proved that the SHCNs are all multiples of each other by a prime, so everything here looks good. I then decided to make an assumption about this function:

f(n) = SHCN(n)

then we can learn a lot by applying this to large numbers. With input 6 the number is 780,296,143,507,862,696,557,498,656,000 = 2^8 * 3^4 * 5^3 * 7^2 * 11^2 * 13 through 61. Since it is known that SHCNs add one more prime at a time, and that HCNs occur at least once for every power of 2, the factorization of one of these shows how frequently SHCNs are compared to all HCNs; the proportion will decrease as the larger primes are added to the end more often than twos are.

The question stuck around for me about the long-term behavior for both series, so I evaluated a few of each for high n, and then wrote a python program to continue checking:

Code: Select all

shcnlist = []

limit = float(raw_input("exponent to use in equation *not too big!* "))
step = float(raw_input("step size to use while searching *not too small!* "))

e = 1

while e < limit&#58;
	
	ceil = 2**e

	
	# taken from primes.py
	n = 2
	primes = &#91;&#93;
	
	
	while n < ceil&#58;
  factors = &#91;&#93;
  m = n
  
  for p in primes&#58;
  	d = float&#40;m&#41;/p
  	if d == int&#40;d&#41;&#58;
    factors.append&#40;p&#41;
    m = m/p
  
  if len&#40;factors&#41; == 0&#58;
  	primes.append&#40;n&#41;
  	factors.append&#40;n&#41;
  else&#58;
  	while m <> 1&#58;
    for p in primes&#58;
    	d = float&#40;m&#41;/p
    	if d == int&#40;d&#41;&#58;
      factors.append&#40;p&#41;
      m = m/p
	
  n = n+1
  

	
	
	shcn = 1
	
	for p in primes&#58;
  shcn = shcn * p**int&#40;1/&#40;p**&#40;float&#40;1&#41;/e&#41;-1&#41;&#41;
  
	if shcnlist.count&#40;shcn&#41; == 0&#58;
  shcnlist.append&#40;shcn&#41;

	
	e = e + step
	
print "...\n...\n..."
print "limit of the real-valued function"
print limit
print "highest non-zero prime exponent"
print ceil
print "partial list of primes"
print primes
print "identified superior highly composite numbers"
print shcnlist
print "consecutive ratios &#40;should be prime if complete&#41;"

not_complete = 0

for n in range&#40;1,len&#40;shcnlist&#41;&#41;&#58;
	print shcnlist&#91;n&#93;/shcnlist&#91;n-1&#93;,
	if primes.count&#40;shcnlist&#91;n&#93;/shcnlist&#91;n-1&#93;&#41; == 0&#58;
  not_complete = 1
  
print "\n"  
  
if not_complete == 1&#58;
	print "NOT",
print "COMPLETE"

print "total amount found"
print len&#40;shcnlist&#41;

for n in shcnlist&#58;
	factors = &#91;&#93;
	m = n
	
	for p in primes&#58;
  d = float&#40;m&#41;/p
  if d == int&#40;d&#41;&#58;
  	factors.append&#40;p&#41;
  	m = m/p
	
	if len&#40;factors&#41; == 0&#58;
  # primes.append&#40;n&#41; *primes have already been found
  factors.append&#40;n&#41;
	else&#58;
  while m <> 1&#58;
  	for p in primes&#58;
    d = float&#40;m&#41;/p
    if d == int&#40;d&#41;&#58;
    	factors.append&#40;p&#41;
    	m = m/p
	factors.sort&#40;&#41;
	print factors

# lcm series

import math

lcmlist = &#91;&#93;
n = 1

while n <= primes&#91;-1&#93;&#58;
	lcm = 1
	
	for p in primes&#58;
  lcm = lcm * p**int&#40;math.log&#40;n&#41;/math.log&#40;p&#41;&#41;
  
	lcmlist.append&#40;lcm&#41;
	n = n + 1

print "lcm of 1 through n"
print lcmlist

print "lcm's which are also shcn's"

for shcn in shcnlist&#58;
	if lcmlist.count&#40;shcn&#41; > 0&#58;
  print shcn
and it found that up to 10^108, the LCM series contains no more SHCNs after 720720. It also shows clearly that the LCM series diverges because it eventually picks up too long of a tail at some point in the millions. The most divisible numbers are not also the smallest to divide the first ones! When I subtract the differences and ask Wolfram&#124;Alpha for a limit as n -> infinity, I am told the limit is 1/2, or, that the relation between a large LCM and a large SHCN is that the LCM has an extra square root of the prime thrown into each exponent, quantifying the long tail.

I wonder what this will do to the notion of some ideal and perfectly divisible huge number...
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icarus
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icarus
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Joined: Apr 11 2006, 12:29 PM

Nov 3 2016, 07:31 PM #2

I wrote about the "El Dorado" of a gigantic but sublimely divisible number base in the thread "Supergrand Bases: Is the magic bullet in the mountains?". (Maybe the dream is akin to the "island" or "continent of stability" regarding the chemical elements...) I've also done some work recently that implies that, generally as numbers increase (we can use primorials to gage "increase") they have not only increasingly vanishingly small numbers of divisors, but slightly less vanishingly small numbers of regulars.

As for your functions, I tried to write an algorithm and am short on time, will have to circle back and look again at what you're saying.
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OctarineBean
OctarineBean

Nov 6 2016, 07:28 PM #3

I wonder in what way the SHCNs and the CANs diverge?

I didn't pursue that because there doesn't seem to be a generator function for those...
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Double sharp
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Nov 23 2016, 12:19 PM #4

I think the island of stability analogy is a little different. The bases past about 16 exist, but they are pretty much unusable by a human society, which can only effectively wield {6, 8, 10, 12, 14, 16} (with 6, 14, and 16 iffy today).

On the other hand, if there were no nuclear shell effects (the basis for the island of stability), elements with more than 103 protons would not be able to exist. So the fact that the seventh row of the table can be completed at all, even if today the half-lives are disappointing (the most being dubnium, 105, with a one-day half-life), is already impressive.

Also, part of the problem is that we do not yet have the methods to get to exactly where the island is supposed to be: it depends not only on proton number but neutron number. Using copernicium (112) as an example, the heaviest isotope we can make today is copernicium-285 (half-life 29 seconds), but if we could stuff in just eight more neutrons, we would get to copernicium-293 (predicted half-life 1200 years). Given that removing eight neutrons gives copernicium-277 (known half-life 690 microseconds), I think the phenomenal growth in half-life is encouraging rather than discouraging.
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icarus
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Apr 13 2018, 07:19 PM #5

Here is a graph that pertains to octarine's 23 Nov 2016 question (see #3 above).

Highly composite and abundant numbers m are products of primorials p_i# (meaning the product of the smallest i primes p). Therefore their prime factors include the smallest ω(m) primes, here "little" omega means the number of distinct prime factors of m. The multiplicities e of these prime factors decrease or hold as p increases. Because of this we might write only the multiplicities of the prime divisors of m = 240 as "4.1.1" as 240 = 2^4 × 3 × 5. Let's call this "multiplicity notation"; we abbreviate it as MN(x).

Now let's consider m = d × d', the product of two divisors, with d' = p_ω(m)#. This is tantamount to writing a multiplicity notation "word" that is a repunit of length ω(m), while the divisor d can be written as MN(m) − 1. For example, the divisors in which we're interested, regarding this graph and vis a vis 240 are 30 × 8, since "4.1.1" → "1.1.1" + "3.0.0" (and of course 3^0 = 1, 5^0 = 1, and we can thus dispense with any trailing zeros in multiplicity notation).

For the graph we need the primitive values of d, thus we compute a sequence S that lists all the d for every HCN and abundant number we know, and take the union of that sequence to arrive at the values that d takes. Therefore we can find the k-th primitive d by writing S(k). The first values in S for HCNs are: {1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, ...} (OEIS A301414).

We plot not the ordered pair (p_ω(m)#, S(k)), but (ω(m), k), and since there are HCNs for each value ω(m) can take, we can just write (n, k) and perhaps use the notation "n: d" to denote a given HCN or abundant number in a compact manner, as representation in any reasonable base quickly becomes a scramble of digits. Thus, 240 would be "3: 3". Here are some {n, k} values for HCNs, with asterisks denoting SHCNs:




Here is a plot with n as the vertical and k as the horizontal, the origin at the upper left. The blue are HCNs and the red abundant numbers. We can see that the abundant numbers and the HCNs have an intersection of 449 terms (these appear in A166981), the largest term there is A002110(69) × A301414(192), which is "69: 10.6.4.3.3.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1".



We can see that the abundant numbers occupy higher n for certain k, but we should also note that A301414 is not sufficient to serve to graph both HCNs and abundant numbers. The largest value of A301414 that is common to both is A301414(314) → "10.5.4.3.2.1.1.1.1.1.1".

Now the SHCN is a special case of HCN, as the "colossally abundant number" is a special case of the abundant number. We can also graph these. The following graph shows the smallest 150 CANs and SHCNs. I've changed the axes so that n increases horizontally toward the right, while k increases vertically downward:


There are 20 terms in the intersection of the SHCNs and CANs (see A224078), and these appear in black. The largest of these twenty is 14: "10.5.4.2.1.1" (decimal: 581442729886633902054768000).

Let's zoom out to a thousand terms of both:


These show the smallest 1000 SHCNs and CANs. Carried to 4000 terms of both, the red and blue "curves" undulate a bit but remain roughly "parallel". (Terms derive from A000705 and A073751).

The divisors d in the k axis are:
d MN(d)
1
2
4
12
24
48
144
720
1440
10080
30240
60480
302400
604800
3326400
6652800
19958400
259459200
518918400
3632428800
61751289600
185253868800
926269344000
1173274502400
3519823507200
17599117536000
35198235072000
809559406656000
1619118813312000
4857356439936000
46954445586048000
140863336758144000
1455587813167488000
1549496704339584000
16011465944842368000
48034397834527104000
96068795669054208000
672481569683379456000
3362407848416897280000
124409090391425199360000
1617318175088527591680000
66310045178629631258880000
2851331942681074144131840000
0
1
2
2.1
3.1
4.1
4.2
4.2.1
5.2.1
5.2.1.1
5.3.1.1
6.3.1.1
6.3.2.1
7.3.2.1
6.3.2.1.1
7.3.2.1.1
7.4.2.1.1
7.4.2.1.1.1
8.4.2.1.1.1
8.4.2.2.1.1
8.4.2.2.1.1.1
8.5.2.2.1.1.1
8.5.3.2.1.1.1
8.4.2.2.1.1.1.1
8.5.2.2.1.1.1.1
8.5.3.2.1.1.1.1
9.5.3.2.1.1.1.1
9.5.3.2.1.1.1.1.1
10.5.3.2.1.1.1.1.1
10.6.3.2.1.1.1.1.1
10.5.3.2.1.1.1.1.1.1
10.6.3.2.1.1.1.1.1.1
10.5.3.2.1.1.1.1.1.1.1
10.6.3.2.2.1.1.1.1.1
10.5.3.2.2.1.1.1.1.1.1
10.6.3.2.2.1.1.1.1.1.1
11.6.3.2.2.1.1.1.1.1.1
11.6.3.3.2.1.1.1.1.1.1
11.6.4.3.2.1.1.1.1.1.1
11.6.4.3.2.1.1.1.1.1.1.1
11.6.4.3.2.2.1.1.1.1.1.1
11.6.4.3.2.2.1.1.1.1.1.1.1
11.6.4.3.2.2.1.1.1.1.1.1.1.1
In this graph we can see where the SHCNs and the CANs diverge. When the SHCNs and CANs share k, the CANs tend to be larger multiples n.

For more, see this study.

Suffice it to say, the number 12 is a superior highly composite and colossally abundant number we can use!
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Double sharp
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Apr 14 2018, 02:06 PM #6

This can be thought of as an illustration of the Strong Law of Small Numbers: "There aren't enough small numbers to meet the many demands made of them." As a result it often happens that the same small number maximises multiple criteria and ends up in all these sequences, which then diverge as the numbers climb up into the stratosphere and beyond. Including the sequence of LCMs of 1 through n as the OP does makes for an even more striking illustration, as the divergence then already happens after 60. The stratospheric metaphor is then very appropriate, given that if we make analogies between human-scale bases and altitudes that can be acclimatised to, the human scale certainly ends well before base 60 (a limit of around 14-20 seems safe enough).

The only numbers in all of these sequences that we can really think about as human-scale bases are 12 and (maybe) 6, and given the "maybe" in front of 6 I think the case is rather clear. ^_^
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Kodegadulo
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Apr 14 2018, 02:47 PM #7

Double sharp wrote:^_^
Does ^_^ = 😄?  Or 😑? Or 😴?
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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icarus
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Joined: Apr 11 2006, 12:29 PM

Apr 14 2018, 09:40 PM #8

I think ^_^ is a pleased face!
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Double sharp
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Apr 15 2018, 04:30 AM #9

Icarus is right. :-)
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icarus
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May 9 2018, 04:26 PM #10

Here is an enlarged poster of the highly composite numbers (A002182) and superior highly composite numbers (A002201), the superabundant numbers (A004394) and colossally abundant numbers (A004490).

We can see that the intersection of SHCNs and CANs (A224078) has just one dozen eight numbers (the dozen is at 2,2, the largest is 739279b90061b4893b0056000z). There are SHCNs that are superabundant (A304234) and colossally abundant numbers that are HCNs (A304235); together with the aforementioned intersection and HCNs that are superabundant, we have A166981, the 4 gross 1 dozen 7 numbers that represent the co-incidence of the two sequences. The largest number that is highly composite and superabundant is 3983226032906342b525149932b15661656639379695969a0993370a82917aba261716b617a3366743887a5a49a523a70816109a87683b82357b17683aab63b32587348b4900000z. Just writing exponents of prime divisors of this number, it is a.6.4.3.3.2.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1. This is not a practical number base, even for aliens from Megamoid, but if it were, the products of the smallest seven dozen ten primes would be regular. Megamoidians would have clean halves, thirds, quarters, fifths, sixths, sevenths, eighths ... 34bz-ths. Downside: memorizing the digits of this base that you do not ever use, nor has anyone ever used in the entire lifetime of the universe. What probably instead happens is when a Megamoidian encounters a number for which a digit never has been used, it has to "cook one up" from primitives that are the prime divisors of this great number. The "hive mind" of the Megamoidians thus inherits that new digit. Most numbers for a Megamoidian are single-digit numbers! But when they encounter a fraction that involves nonterminating expansion, only one sig-fig is required and even that is too darn exact. Thankfully Megamoidians are far from earth because earthlings are confounding, with those small shallow bases, rows of monotonous digits the Megamoidian mind cannot grok or distinguish.

We see that A301414 is sufficient to represent both HCNs and superabundant numbers up to A301414(272z) = 19071140072486500000z = 2^a × 3⁵ × 5⁴ × 7³ × b² × 11 × 15 × 17 × 1b × 25 × 27. We need a new sequence to represent both HCNs and superabundant numbers.



I am not sure these things are of interest to people here anymore. Maybe you like to think of immense numbers.

The question had derived from my thinking that superior highly composite numbers (SHCNs) were "ideal bases": {2, 6, 12, 60, 120, 360, 2520, 5040, ...} these shown in black. I could not really distinguish SHCNs from colossally abundant numbers; not that the difference is sensible among small numbers. Twelve is part of this sequence and so is 60 and another favorite, 120. Numbers like 360 and 2520 don't quite work as nicely as 12, 60, and 120 (the latter is really groovy and we have known a few who do love it greatly). The problem with numbers greater than 12 in this series is that they are just way large and some method of taming them is required to employ them. The cost of taming these numbers seems to vastly outweigh the very reason for going to them. The number of divisors and even regular numbers for SHCNs becomes vanishingly small when compared to the number of digits (numbers m < n, n the base), and if the range of digits is great enough never to really use all of them, one wonders what use or purpose there could be for such a large grouping.

This study is like going around the world when all I wanted was to see if I was right about the corner store. I am happy I went on it, and it seems to reveal pictorially a relationship between primorials, SHCNs, and colosally abundant numbers, etc. But it probably no longer harbors any lesson we'd seek about number bases, save that the "magic bullet does not lie in the mountains".

Regarding number bases, for divisibility and flexibility, it would seem sufficient that we would need only consider the smallest handful of black pixels!
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