I like highly composite numbers!
https://mathworld.wolfram.com/HighlyCompositeNumber.html
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So in the very first line, it says
[quote]Highly composite numbers are numbers such thatdivisor function[i]d(n) = σ0(n)[/i](i.e., the number of divisors of[i]n[/i]) is greater than for any smaller[i]n[/i].[/quote]
It seems to be written by an ESL speaker...but ok, got the idea.
So now that you have led me to this, please further inform me regarding why you "like highly composite numbers"?
[quote]Highly composite numbers are numbers such thatdivisor function[i]d(n) = σ0(n)[/i](i.e., the number of divisors of[i]n[/i]) is greater than for any smaller[i]n[/i].[/quote]
It seems to be written by an ESL speaker...but ok, got the idea.
So now that you have led me to this, please further inform me regarding why you "like highly composite numbers"?
SomeMichGuy · M
@ElwoodBlues I really would like to thank the Academy of Best Comments for selecting my reply. With such a robust field of competitors, it was an honor to be amongst such fine answerers. To be actually selected was furthest from my mind, and it is a career high for me.
😉
Seriously, where are the people engaging in this?
😉
Seriously, where are the people engaging in this?
ElwoodBlues · M
@SomeMichGuy Well, it's like this. I joined SW back in late June or early July. And at first things were OK, but then my posts seemed to go invisible. Literally. Only I could see what I posted. But it took me a while to realize that.
Anyway, somewhere in that process, I was trying posting in different subject areas to see if that made a difference, and I read a list of newly made groups - groups that were hierarchical and didn't start with "I <verb> ..." So I tried posting in the new math group.
I believe this post is the first post in the group. Now, as to why, ... I'll do my best, but it might not make much sense. I was a physics major in college, so very accustomed to the real number line, complex numbers, logs, trig, etc. I would also, on a car trip or bus trip, occupy my mind sometimes by factoring numbers that I saw - a 4 digit license plate number is usually my limit. And it was fun finding little mental tricks to test for divisibility without the bother of long division.
You keep factoring until you're down to the prime factors, and primes are pretty cool, there's an easy proof that there is an infinite number of primes; large primes are part of the magic of public key encryption, etc. Anyway, one day about 10 years ago I was going down some rabbit hole on the web and I was reading about Ramanujan and I came across the sequence of highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840 ...
Each successive member in the sequence is the lowest number with more divisors than the previous member. They are like the [i]opposite[/i] of primes. I thought that was just so cool. And because primes are irregularly spaced and (so far for us humans) unpredictable, that means the highly composites are also ultimately unpredictable (I think.) Wow, this is getting long, but I'm enjoying setting it down.
I remember going to England as a kid with my brother and parents, back in the days of shillings etc. My parents had trouble with the coinage, but it was no prob for us kids. And I remember thinking (maybe not on that trip) that it was cool that if 3 kids found a pound note, they could divide it equally. Or 4 or 5 or 6,8,10,12, etc.
There were 240 old pence per pound - highly composite. 360 degrees in a circle - highly composite. 7 factorial = 5040 - highly composite (I think 7 factorial might be the highest factorial in the list). To me they're just a fun group of numbers.
P.S.: about doing mental calculations. Some people are human calculators; not me. I think I read that the author Vladimir Nabokov, as a child, had a sort of calculator in his head that would run autonomously; that he could give it a problem, go on about his activities, and hours or days later it would produce an answer. Anyway, that's not me. I have to do it very consciously. If I want to see if 377 is divisible by 13, I'll start with a nearby problem, like I'll I multiply 13 by 30, getting 390, easy mental math. Then I subtract 13 from 390, first taking away 10, then 3 more, and lo and behold 377 factors into 13X29, no long division necessary, all easy mental arithmetic!
P.P.S.: did you know that 3 times 7 times 37 equals 777. Kinda makes ya think! Actually 777 doesn't really make me think. I have no patience for numerology, I just think numbers are fun and amusing. Here's the story of 1729, the Hardy - Ramanujan number. https://www.businesstoday.in/latest/trends/story/national-mathematics-day-why-is-1729-special-magic-of-hardy-ramanujan-number-282270-2020-12-22
Anyway, somewhere in that process, I was trying posting in different subject areas to see if that made a difference, and I read a list of newly made groups - groups that were hierarchical and didn't start with "I <verb> ..." So I tried posting in the new math group.
I believe this post is the first post in the group. Now, as to why, ... I'll do my best, but it might not make much sense. I was a physics major in college, so very accustomed to the real number line, complex numbers, logs, trig, etc. I would also, on a car trip or bus trip, occupy my mind sometimes by factoring numbers that I saw - a 4 digit license plate number is usually my limit. And it was fun finding little mental tricks to test for divisibility without the bother of long division.
You keep factoring until you're down to the prime factors, and primes are pretty cool, there's an easy proof that there is an infinite number of primes; large primes are part of the magic of public key encryption, etc. Anyway, one day about 10 years ago I was going down some rabbit hole on the web and I was reading about Ramanujan and I came across the sequence of highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840 ...
Each successive member in the sequence is the lowest number with more divisors than the previous member. They are like the [i]opposite[/i] of primes. I thought that was just so cool. And because primes are irregularly spaced and (so far for us humans) unpredictable, that means the highly composites are also ultimately unpredictable (I think.) Wow, this is getting long, but I'm enjoying setting it down.
I remember going to England as a kid with my brother and parents, back in the days of shillings etc. My parents had trouble with the coinage, but it was no prob for us kids. And I remember thinking (maybe not on that trip) that it was cool that if 3 kids found a pound note, they could divide it equally. Or 4 or 5 or 6,8,10,12, etc.
There were 240 old pence per pound - highly composite. 360 degrees in a circle - highly composite. 7 factorial = 5040 - highly composite (I think 7 factorial might be the highest factorial in the list). To me they're just a fun group of numbers.
P.S.: about doing mental calculations. Some people are human calculators; not me. I think I read that the author Vladimir Nabokov, as a child, had a sort of calculator in his head that would run autonomously; that he could give it a problem, go on about his activities, and hours or days later it would produce an answer. Anyway, that's not me. I have to do it very consciously. If I want to see if 377 is divisible by 13, I'll start with a nearby problem, like I'll I multiply 13 by 30, getting 390, easy mental math. Then I subtract 13 from 390, first taking away 10, then 3 more, and lo and behold 377 factors into 13X29, no long division necessary, all easy mental arithmetic!
P.P.S.: did you know that 3 times 7 times 37 equals 777. Kinda makes ya think! Actually 777 doesn't really make me think. I have no patience for numerology, I just think numbers are fun and amusing. Here's the story of 1729, the Hardy - Ramanujan number. https://www.businesstoday.in/latest/trends/story/national-mathematics-day-why-is-1729-special-magic-of-hardy-ramanujan-number-282270-2020-12-22
SomeMichGuy · M
@ElwoodBlues Thank you!
I also studied Physics (and EE and...) and am interested in learning more of many things, including math.
Number theory is somewhat like philosophy: very accessible, but you can dive deep and get very technical, beyond the depth of a dilettante.
The notion is interesting and I, too, was aware of the old English pence / shilling structure--through reading, then, not travel--and how odd it seemed as a base... But, of course, quite practical in the way you mention!
[Have you read about Babylonian mathematics? Base 60, source of our 360° circle and associated time units, and they used sawtooth approximations to sinusoidal functions to predict the locations of the planets, etc. Check out this wikipedia article: [i]https://en.m.wikipedia.org/wiki/Sexagesimal[/i])
I also studied Physics (and EE and...) and am interested in learning more of many things, including math.
Number theory is somewhat like philosophy: very accessible, but you can dive deep and get very technical, beyond the depth of a dilettante.
The notion is interesting and I, too, was aware of the old English pence / shilling structure--through reading, then, not travel--and how odd it seemed as a base... But, of course, quite practical in the way you mention!
[Have you read about Babylonian mathematics? Base 60, source of our 360° circle and associated time units, and they used sawtooth approximations to sinusoidal functions to predict the locations of the planets, etc. Check out this wikipedia article: [i]https://en.m.wikipedia.org/wiki/Sexagesimal[/i])
ElwoodBlues · M
SomeMichGuy · M
@ElwoodBlues Sorry you didn't get Best Comment; it was a tough field of contenders. 😆😆😆
ElwoodBlues · M
@SomeMichGuy It's up to the arbitrary whim original poster to select or not select a best comment. But there was no BC click on my own comment; I guess the OP can't select self. Congrats again on your victory!!
SomeMichGuy · M
@ElwoodBlues lol Thank you, & I know, but the absence of real discussion here made me want to make fun of the fact that there was NO competition.