Note: Some of you are already doing something like what is discussed here.
You know who you are. But read on anyway.
Do you have a problem with passwords? It seems that a lot of people do but it doesn’t have to be that way. I mean, just write them down and keep the list in places where you can find it. For example, write them in a notebook; Or on a piece of paper and keep it in your wallet; Or put a copy of the list on Google Keep; Or write them on PostIts and stick them on your monitor. But, you ask, what happens if someone gets a copy somehow?; like my ex getting a copy of my Google Keep notes! No problem. What you need is the world’s best password manager. And it’s FREE! And I’m going to give it to you now!
This “password manager” is just a list of 26 words that you must memorize where each word starts with one of the letters of the alphabet (az)… And… a simple algorithm that you apply. For example…
My list of 26 words is the military alphabet (Alpha, Bravo, Charlie, … Zulu which I have memorized). Of course, it could just as well be a list of names like Anna, Brittany, Chuck, … Zorro. But I will use the military alphabet. In addition, you will need a simple algorithm to use for numbers and special characters. Here’s how it works…
Let’s say I need a new password for the Bank Of America. Let’s say it needs at least 8 characters, at least 1 uppercase letter, at least 1 number, and at least 1 special character. So I might write down the following in my notebook (or on a postit, etc.):
BofA12345&@ <=== This is my password that I write down somewhere
But on the Bank of America web site when I type in my new password I would enter the following:
BrosfoAl29@ <===== This is what I type in the password field on browser
The next time I log in to the BofA site I will look at my Postit Note that says “Bank of America password is “BofA12345&@” but I will apply my password “translation rules” to this and instead enter “BrosfoAl29@”
Here are my rules…
Letters
Using the military alphabet I will substitute the first 2 letters of the military word to replace each corresponding letter on my Postit Note.
If the letter on the Postit is uppercase I will also make the first of these 2 letters of the military word uppercase as well.
Note that the first letter of the pretranslated password on the Postit Note is uppercase B. B stands for bravo. So in place of the “B” on the Postit I will key in “Br” in the password field on the browser. Note that since the B of the writtendown password is uppercase then the first of the 2 replacement letters will also be uppercase. Hence the “B” from the “password” on the postit gets replaced by “Br” when I actually key it in.
For lowercase letters on the writtendown postit “password” I will simply use the first 2 letters of the corresponding military word. So the military word for “o” is oscar. Using the first 2 letters of oscar yields “os” (both replacement letters will be lowercase since the “o” in the writtendown BofA12345&@ is lowercase.
Numbers
Numbers written in the “postit password” are strings of 3 or more digits. For example 123, 935, 1234, 12345, and so on. Here’s the substitution/translation rule…
Multiply all but the last number to get a product then add the last number to that result. For example, 123 from the written postit password becomes (1×2) + 3 which yields 5 so I would enter 5 in the browser. Let’s use 12345. This becomes (1x2x3x4) + 5 which yields 29. So “29” would be entered in the browser. How about 321? This gets translated to (3×2) + 1 which yields 7 and it’s the “7” that gets entered into the browser.
Special Characters
When I see a “&” I simply enter the special character that follows. So “&@” simply becomes “@”
In a nutshell, you should come up with something simple yet meaningful as your “written down password.” When you are coming up with the writtendown password you will be keeping in mind and considering your translation rules so keep it simple and meaningful. For example, you might use BillG456 for your Microsoft email password. You write down “My password is BillG456” which is somewhat meaningful to you. But when you actually enter your password in the browser you would enter “BrinliliGo26” which you can think of as
Bravo india lima lima Golf ((4×5)+6)= 26
where only the highlighted characters are entered.
The above is merely an example algorithm. You might use a similar set of rules based on the names of people you know for the letters. And you can come up with your own simple scheme for numbers or special characters. Or, just use the above asis. But whatever you do, the point is that you “write down” your “pretranslated” passwords on paper, and/or put them in a text file, or write them in Google Keep, or whatever is convenient for you!!! . And whenever you actually enter them on the computer you will do the translation in your head as you type them.
And if someone gets hold of your “paper” password list do you think they would be able to figure out what it’s all about. Would they know or even think that some translation process is needed? Would they have any clue what the translation process is? Of course not! All of that information is perfectly SAFE IN YOUR HEAD!
Trust me… this is really easy and you will get ridiculously efficient at translating, on the fly, your paper passwords to what you key in. I’ve been using this method for over 20 years! This method is so foolproof and secure that I even write my pretranslated “paper” passwords on the front of my credit cards!
Trust me! This is Middle School level stuff! You can do this or something similar that you come up with. You can have the absolute best password manager ever. And you will carry it with you wherever you go. And it doesn’t cost anything because IT’S YOU! IT’S IN YOUR HEAD! And you can have fun doing this too! Trust me… you’ll be amazed at how easy and fast it becomes after just a short time. You will soon be making the translations as you type while barely thinking about it. It’s like riding a bike.
If you don’t think you can do this then you need therapy. Here’s my therapist. She’s good.
Moving on… once again, you don’t have to use the above “rules.” Play with the idea for a couple of weeks and then come up with your own translation rules. You’ll have fun. And best of all…
Once you start using your new password manager
you’ll find that
managing your passwords is as simple as just writing them down.
The End
]]>This is Part 2 about the MSNBC “documentary” on Robert Mueller (and about trust). Part 1 is required reading in order to appreciate what’s going on here.
Click HERE to review/read Part 1
Also, as with Part 1, this post is not about Trump. That said, if at any point you find yourself saying or thinking the word “Trump” then… well, you know the drill if you read Part 1.
Review of Part 1
The following information comes from the articles and videos linked to in part 1. You read/watched them, right?
We learned that MSNBC recently produced and aired a “documentary” about Robert Mueller. And we learned that Robert Mueller had been appointed the FBI Director shortly before 9/11. And that just a few weeks later there were the anthrax attacks.
Mueller personally took control of and personally managed the anthrax investigation and after a short time came to the conclusion that the anthrax “culprit” was one Steven Hatfill. After a long intensive investigation (which lasted 7 years!), and after destroying Mr. Hatfill’s life, a Federal court exonerated Hatfill and also determined that Hatfill should never have even been a “person of interest.” The Justice Department then settled with Hatfill and paid him $5.8 million for the suffering it caused him.
In addition, the FBI (and Mueller) had early on been warned and informed about the actual culprit who could have easily been captured immediately if the investigators (under Mueller’s personal tutelage) had simply “followed the access.” You don’t buy anthrax at your corner drug store! There are very few places where anthrax is kept and the stuff is locked up tight behind strong doors with military guards and all access is meticulously documented! Even you or me could have easily tracked down the guilty party! But Mueller fucked up big time because he thought he was smarter than everyone else and instead “followed the bloodhounds” (quite literally! Read/view the articles/videos documented in Part 1).
So Mueller bungled the biggest case the FBI had ever seen. And he bungled it for 7 years! Until finally a Judge had to step in and put a halt to the travesty that Mueller had been “managing” for 7 years!
But this post is NOT about Robert Mueller…
So now we skip forward about 10 years to today where Mueller is in the news as a “special counsel” investigating President Trump. But again, this story is not about Robert Mueller; And again, neither is it about Trump.
It’s about trust
“And now for the rest of the story…”
Today… 10 years later MSNBC produces a “documentary” (BioPic) about Mueller. You watched it when you read Part 1. You did watch it, didn’t you? Anyway, nowhere in that “documentary” was anything mentioned about Mueller’s Hatfill/anthrax case which he personally managed (bungled?) for 7 years until the courts had to step in to tell him to let it go! Did I mention it was one of the biggest and most important FBI cases ever? Did I mention that Mueller personally managed that case? Did I mention it went on for 7 years!?
Oh, I almost forgot. Did I mention that there is
no reference to any of this the MSNBC Mueller “documentary?”
I’m going to go way out on a limb and state that this “omission” was/is intentional. And I think it’s a safe bet that the omission was not due to some intern. Multiple people within MSNBC had to agree to not mention this extremely important 7 year period and the related events of Mueller’s life. And I’ll bet they’re high up in the MSNBC food chain.
So who are they? Beats me. And why did they do this? Don’t know(but we both can speculate). But I bet Rachel could figure it out. Send her a tweet… after enough of them maybe she’ll accept the challenge. Anyway, when all is said and done, will you now be at least Skeptical (with a capital S) about what appears on MSNBC? And after you think about all this for a while, what will your level of trust be visavis MSNBC? Take your time before deciding. I’m sure they took their time in deciding what you can’t see.
Now It’s The End Of The Story
]]>most wonderful human being I’ve ever known in my life.”
This post is not about Trump. You’ll find out what it’s about a little later. That said, if at any point you find yourself saying or thinking the word “Trump” then please stop and tell yourself
“Focus. If I just focus. I know I can do this. I can have a life if I just stop thinking about it! All I have to do is focus!”
Then continue. You can do this; You know you can.
Now we can begin… Last Night MSNBC aired its “documentary/Bio” of Robert Mueller. Here is a link to a copy that someone put on YouTube. It’s unknown how long it will remain due to “copyright” issues. I could not find any official copies on MSNBC.com. If I do find such copies I will update this post. That said, I encourage you to watch this MSNBC “documentary” in its entirety.
Headliners: MSNBC “documentary/bio” on Robert Mueller – Aired 4/15/2018 (about 35 minutes)
https://www.youtube.com/watch?v=tOuESoEFigw#t=0s
If you’re too lazy and don’t want to spend the time then watch the short version of the MSNBC “documentary” just below:
With the above in mind, follow the links way below and read the articles and watch the videos. They are about the Steven Hatfill case; that is, it’s about the anthrax attacks in the U.S. after 9/11 and the FBI’s associated investigation in which an innocent man was targeted, and his life ruined, while the actual perpetrator was quite literally under their noses but was ignored even though he should have been the obvious and primary suspect if they simply would have followed the trail of those few who had access to the anthrax! And whose access was fully documented! And who others had warned the FBI about!
The Hatfill case was arguably the biggest and most important case in FBI history. It went on for years until… you’ll find out why. And interestingly enough, the man in charge of the case and in charge of the FBI’s investigation was… you guessed it… Raymond Sh… I mean, Robert Mueller.
Down below you’ll find a list of links to a wide variety of articles from a wide variety of sources that discuss the Hatfill case and Robert Mueller’s association with it. I encourage you to take the time to read and watch and learn about something you lived thru but which you have probably forgotten. And learn how someone at the forefront of today’s events was in charge of and intimately involved in the Hatfill/anthrax case which ended oh so very badly. That “someone” was/is Robert Mueller.
Here is the BIG question. Did you watch the MSNBC biopic about Mueller (aka Raymond Shaw in the parlance of MSNBC)? In the MSNBC “documentary” did you see any reference to Mueller’s association with the Hatfill/anthrax debacle? No? Hmmmm… that’s odd. I wonder why not? A mere accidental oversight maybe?
Is it any wonder that people do not trust what is reported in the “media?”; or in this case, what is NOT reported!
Here is the list of links to articles and videos on the Hatfill case from a wide variety of sources. Btw, there are many other sources of information. I am sure you can easily find them.
The Atlantic Magazine – 2010
https://www.theatlantic.com/magazine/archive/2010/05/thewrongman/308019/
Huffington Post
https://www.huffingtonpost.com/entry/conflictsofinterestandethicsrobertmuellerand_us_5936a148e4b033940169cdc8
L.A. Times – 2017
http://www.latimes.com/politics/lanapolmuellerrecord20171122story.html
NY Times – 2008
https://www.nytimes.com/2008/06/28/washington/28hatfill.html
Orange County Register – 2017
https://www.ocregister.com/2017/05/21/comeymuellerbungledbiganthraxcasetogether/
The Federalist – February 2018
http://thefederalist.com/2018/02/08/robertmuellerbotchinginvestigationssinceanthraxattacks/
mark levin
https://www.youtube.com/watch?v=qDMN15LpXhA
Associated Press – Aug 1, 2008
https://www.youtube.com/watch?v=4slGYVsI_es
Associated Press – Aug 4, 2008
https://www.youtube.com/watch?v=a586kg5vZUA
CNN death by mail – 2011
https://www.youtube.com/watch?v=fuZi52lZVuw
MSNBC hatfill lawsuit – exonnerated 2008
https://www.youtube.com/watch?v=UNztslM4NZ0
NBC Today Show – 2014 hatfill wrongly pursued by FBI
https://www.youtube.com/watch?v=v7n6bVrQOlE
NBC news 2015
https://www.youtube.com/watch?v=FG95QvnReNA
https://www.congress.gov/congressionalrecord/2017/6/12/housesection/article/H48481
The End
]]>The “complaint” sometimes (often?) heard is that a major factor in the student loan debt crisis is the “huge” costs due to tuition. More to the point, a major “complaint” is that tuition costs are astronomically high resulting in crushing student debt just to pay the cost (of tuition). The question, visavis the discussion we were having is whether tuition is a major factor; or more importantly, must the cost of tuition necessarily result high student debt?
Florida State Univ. Tuition
InState $4,640/year ($18,600 for 4 years). I’d say that’s pretty damned good!
However, if one goes to a community college for the first 2 years then the total tuition over 4 years will be about $13,000. In other words, graduate from Florida State for about $13,000 in tuition.
UCLA and all other UC system schools
InState $12,600/yr about $50,000 for 4 years. That’s a whole lot more expensive than when I attended and $50k is significant. BUT…
Again, however, if one goes to a California community college for the first 2 years then the total tuition for those 2 years is only $2208. Or
graduate from UCLA for a total tuition of $27,408 (quite a savings when compared to $50k).
UCLA has shattered its own record as the nation’s most popular college choice for high school seniors, attracting more than 113,000 freshman applications for fall 2018, according to preliminary data released Thursday.
http://www.latimes.com/local/education/highered/lameeduuclaapplicants20171214story.html
California State Universities
If one decides to instead attend a California State University (as opposed to a UC system school) then the tuition is only $5742/year or about $22,000 for all 4 years! Or, and again, one could attend a community college for the first 2 years then the tuition cost for all 4 years totals out to about $13,700. That’s pretty damned cheap to attend one of the many excellent universities including CalPoly, San Jose, San Diego, etc.
The Takeaway
One can, if they want, pay enormous amounts in tuition and they can finance it with student loans; but they don’t have to. The cost of tuition does NOT have to result in crushing debt in order to obtain an excellent college education.
Stop making yourself peanut butter sandwiches.
The End
]]>
NYT “TRUTH” ad follows
At any rate, the NYT “truth” ads, like the CNN “apple ads,” implore you to trust them. They implore you to believe that they tell you the truth… that they wouldn’t lie to you… all of which tells you… well, you decide.
So what does it really mean when a major media outlet begs us to believe that they are telling us the truth? I mean really, what is that all about?
Anyway, i find it intriguing that now there are 2 major media outlets that are begging for us to believe them. What’s THAT about?
“And that’s the way it is. Trust me. No, really, trust me.”
The End
]]>
Take a look…
Ballerina
Penguin
The “Ballerina” video is a PSA about drunk driving and the subject is given serious treatment. Oddly enough the “Ballerina” PSA is often followed immediately by the “Texting Penguin” PSA.
The Texting Penguin is a playful, joking, and amusing treatment of texting while driving (don’t forget to listen to the accompanying music). Put another way, it’s a lighthearted treatment of behavior that KILLS 11 TEENAGERS EVERY DAY in the U.S. (and kills adults and preteens too). But judging by the video one would think that texting while driving is a minor inconvenience at most (running into a snow bank… c’mon) and mostly a funny thing that happens (did you see the surprised look on the Penguin’s face and listen to the lighthearted music when it saw the snow bank?… how cute is that!).
And all that, of course, brings us to the cable/network “news” nonstop topic du jour… schools, armed guards, failed background checks, 17 victims, Gun Control issues, ….. A serious and important topic to be sure. Teenagers killed at school… It’s infinitely more important and the priority is infinitely higher than a cute Penguin who for many years has been killing 11 kids per day.
“Be careful out there.”
But Wait! There’s More!
After corresponding with a reader I realized I needed to add a little clarity to the point I am trying to make…
The point I was trying to make is that the Penguin psa, because of the nonserious manner/tone in which it was done, implies that texting and driving is only a very minor problem at best. However, when we actually look at the stats we find it’s a way bigger problem than even school shootings but that it’s essentially ignored.
11 Dead
and 800 SERIOUS injuries EACH DAY!
And yet the psa is, how should i say it?… “Jovial.” Am i the only one who sees this?
Anyway, it’s a VERY lowpriority (nopriority?) public issue.
What are they trying to say? Who are they trying to convince?; is it you (or me)? Why? Do they think that you (or me) can’t figure out what the truth is? Come on in…
Lock The Gates!
In the first video (below)… Are they trying to point the metaphorical finger at “the enemies of truth”… or… are they pathetically pleading for you (me) to believe that CNN tells the truth? In either case, why do they feel the need to convince the viewer? Really…
Why does CNN think they need to run an ad to convince the viewer that they are telling the truth?
WHY?
In the second video (below)… I particularly like the part where the narrator says
“over and over and over again”
Btw, look on YouTube where you’ll find tons of interesting and clever variations on these CNN “ads.”
The End
Breaking News…
Got a response on the above to which I responded…
Chris said…
Those are shitty commentary/commercials….but you DO know
why they think they have to defend accuracy in
media…specifically when some elected official keeps saying they,
CNN SPECIFICALLY, are fake news?
Sometimes I forget how old we are and how much of history we actually
had to learn and LIVED through…not true for those 30s and younger …. If
it’s more than five years ago…it didn’t happen.
To which I responded…
“And that’s the way it is. Trust me. No, really, trust me.”
Perhaps I should add all the above to the post.
Chris responded…
No Rebuttal needed
Then I said…
OK. Now it’s THE END
Open The Gates!
]]>
This post is a kind of “stream of consciousness” epilogue to the Math Mystery Project; aka the Harmonic Series of Composites project. It’s a “story” of sorts; an additional chronicling of the “ending” of the project. Anyway, none of what follows will make any sense at all until/unless you’ve read the first 3 “parts” of the “Math Mystery” story. Here’s the links if you want to get started.
1 https://itbegsthequestion.com/ anactualmathmystery/
2 https://itbegsthequestion.com/ mathmysteryfollowup/
3 https://itbegsthequestion.com/ mathmysterywrapup/
Ok, we’re back. As you probably noticed, this “project” ended up being something completely different than what it started out as. That’s not uncommon either with the projects or with life in general (duh). We start out investigating something, notice something “odd” or interesting, or shiny, and then it’s off in another direction; sometimes a bunny trail into the weeds but sometimes something meaningful and fascinating and worth remembering.
Anyway, at the end of Part 3 of the Math Mystery, I decided to find and email some professional mathematicians to see what they thought of what was “discovered.” I’ve not heard back from the first 2 yet but here is the email I sent to them (btw I am including this “first” email and parts of some others and the responses in order to document what was being thought at the time…. a personal historical log. Ooh! A “phlog!” Anyway, here’s the first email to two pros:
Dear Mr. Roelandts,
I am hoping you can solve a problem for me (or direct me to a doc that will).
Then it’s my conjecture that:
ln(π(n)) + Sp ~ Sn
In words:
I also copied the above email to a friend of mine in California (JGM) and here is what transpired:
Gerry – I sent your email to a mathematics friend and this is what he said
“looks like a number theory cleverness of Ramanujan (encouraged by Hardy)”
Note… emphasis is mine above.
To which I responded
Then I googled “cleverness of Ramanujan” as mentioned above and found out what they were (presumably) talking about so I wrote back to JGM.
John,
Thanks to google I now get the “cleverness of Ramanujan” reference which I’ll take as a compliment though (largely?) maybe undeserved. Here is a link
https://www.wired.com/2016/04/whowasramanujan/
The wired magazine article linked to just above contains the following (emphasis below is mine).
“Again, they began unpromisingly, with rather vague statements about having a method to count the number of primes up to a given size. But by page 3, there were definite formulas for sums and integrals and things. Some of them looked at least from a distance like the kinds of things that were, for example, in Hardy’s papers. But some were definitely more exotic. Their general texture, though, was typical of these types of math formulas. But many of the actual formulas were quite surprising—often claiming that things one wouldn’t expect to be related at all were actually mathematically equal.“
btw, the Wired Magazine article referenced above was written by Stephen Wolfram. You may already know about him but if not, just google him (and prepare to be amazed).
btwbtw with regard to “often claiming that things one wouldn’t expect to be related at all were actually mathematically equal” ……. It has VERY often been my experience that what often shines a bright light on relationships of “number stuff” are the results of actually doing computations… BILLIONS and BILLIONS of computations (we need to write programs for this!). Then analyzing and comparing the results (the numbers) using things like logs (which often plays a role in number theory) and using a good scientific calculator ( like the one that comes with Windows, or Google’s (just type “calculator” in the address bar of the Chrome browser)). All this just to help us “see” relationships. Of course, we need to have a feel for what we may want to compare and then we need to actually write the code to compute that stuff. Then the results will hint that we need some additional result which we dutifully add to the code… rinse and repeat. Of course, not every number computed will yield interesting results so we need to be prepared to “waste” many hours of effort writing and revising these programs. And keep in mind… we don’t get paid for this shit.
btwbtwbtw… the mathematicians of the past would very often do the same thing (perform gazillions of computations) and for the same reasons; but they did it by hand!… when paper was not easy to come by! Can you imagine what THEY would have thought, and given, to have a computer!?
Time to go… the black escalade is back.
gerry
So the above email chain is what I was thinking at the time. Then about a week later while googleing “harmonic series of composites” and related terms, I ended up on the OEIS web site and finding its founder, Neil Sloane.
IMPORTANT! Follow these links
to learn about OEIS and Neil Sloane:
So who’s Neil? And what is OEIS?
Science News https://www.sciencenews.org/article/patterncollector
Wired Magazine Meet the Guy Who Sorts All the World’s Numbers in His Attic
For OEIS site, see —>> https://oeis.org/
and for my “contribution” to humanity see
https://oeis.org/search?q=A296358&language=english&go=Search
So I emailed Mr. Sloane to see what a recognized expert had to say about all this Harmonic Series of Composites stuff. The email and his response follows:
Dear Mr. Sloane,
I am hoping you can help to solve a (simple?) series related problem for me (or direct me to a document that will). I can’t remember how I arrived at your name as a likely candidate for helping but it (of course) involved numerous google searches that eventually landed me on OEIS and one of its pages:
https://oeis.org/wiki/Talk: Harmonic_series_of_the_ composites
That page did not help directly but with just a few clicks around OEIS I stumbled on you. Anyway, with all that said, here is the short story… i.e. “Just the problem” as Sgt Joe Friday would say.
The two most “famous” series are the “vanilla” Harmonic Series, and of course, the Harmonic Series of Primes. The shorthand/symbols I use for these series is S_{n} and S_{p} (S for Series, n for natural numbers, and p for primes). When we go to infinity the
vanilla Harmonic Series ~ Ln(N)
and the Harmonic Series of Primes ~ Ln(Ln(N))
or…
S_{n} ~ Ln(N)
S_{p} ~ Ln(Sn) or, S_{p} ~ Ln(Ln(N))
We see the above in a lot of places.
But what about the third leg of the Harmonic Series “Triad?” What about the Harmonic Series of Composites ( we’ll call it S_{c} )? We find references and extensive literature all over the internet about the vanilla Harmonic Series (S_{n} ) and about the Harmonic Series of Primes (Sp) but the number of references to the Harmonic Series of Composites is just about nil! Why? Especially when its “value” is so interesting!!! And this brings us to the main points of this whole discussion.
1. What is the value of S_{c} ?
2. Why?
So what is its value? Of course, when we take all of the primes (p) away from the natural numbers (n) we are left with just composites.
S_{c} = S_{n} – S_{p}
Let’s massage the above a little and we’ll get…
S_{n} = Ln(N) and S_{p} = Ln(Ln(N))
so…
S_{c} = Ln(N) – Ln(Ln(N))
Here is where the magic happens with a little log arithmetic
S_{c} = Ln( N / Ln(N))
AND Note that … N / Ln(N) is the prime counting function π(N) so…
S_{c} = Ln(π(N))
or, said in words,
The Harmonic Series of Composites (aka S_{c} )
approaches the log of the number of primes in N as
N → infinity!
Note that experimental results (computations) support the above at least thru N = 1 billion. You will see this (and more) if you read the related blog articles (links are further down).
With all that said, some questions… Did I miss something? Did I make some boneheaded mistake along the way?
Why don’t I find any literature about any of this? And, do you know of any literature/documents that cover the Harmonic Series of Composites?
I am Especially looking for an “intuitive” reason for WHY the log of the count of primes should have anything to do with the value of the Harmonic Series of Composites.
Thanks for any insights or other information you may have.
Gerry
P.S. Below are 3 links to a 3part “story”/ history about all of this on my blog if you are interested. And once again, thanks for any insights you might have on the above.
1 https://itbegsthequestion.com/ anactualmathmystery/
2 https://itbegsthequestion.com/ mathmysteryfollowup/
3 https://itbegsthequestion.com/ mathmysterywrapup/
Mr. Sloane replied to the above email with this response (again, emphasis is mine).
Dear Gerry,
I agree with your estimate for Sum_{k=1..n} 1/composite(k).
Although the OEIS has many similar sequences of fractions – here is a list:
Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512 –
it did not have that one. We did have the successive numerators (A282511)
but not the denominators, so I created A296358 for them (actually the denominators of Sum 1/nonprimes, A282512, are essentially the same sequence). So Sum 1/composite(n) is now A282511/A296358. I then added your interesting comment about the asymptotics to A296358. Although it is easy to prove, it is a nice observation, and I had not seen it before.
Thanks for writing.
Neil Sloane
Easy and simple are big pluses in math. And, the man who has seen it all in the world of integer sequences hadn’t seen this one! So it turns out that I was not crazy!! And he agrees with my proof. But… But… he did not hazard a guess as to WHY π(N), or more specifically, Ln(π(N)) should play a role in the value of the Harmonic Series of Composites… But it does… and it is! Once again, it appears to be another case of what was said in the Wired Magazine article about Ramanujan:
“things one wouldn’t expect to be related at all were actually mathematically equal.”
So Mr. Sloane created my own entry/page in OEIS. It’s
http://oeis.org/search?q=A296358&language=english&go=Search
As said in the Guardian article
“It is also a badge of honour to get a sequence accepted – although Neil and the administrators tend to be very generous in their selection criteria. Still, you need to think up something that hasn’t been thought up before!”
and…
“The OEIS is also a source of interesting problems, quite often that arise from the recreational mathematics community. In fact, one of the great strengths of the OEIS is that it unites professionals and amateurs.”
Anyway… The “Mathematical equivalence” discovered during this project was and IS VERY surprising and unexpected.
S_{c} = Ln(π(N))
And it’s amazingly simple and easy to prove as Mr. Sloane will attest to. And one of the biggest surprises (and mysteries) is that, as far as I can tell, it’s been essentially unknown!… Even to Mr. Sloane! And just as surprising is that it all started with some unexplained notes on a wrinkled piece of notebook paper stuffed into a remote corner of an IKEA bookshelf! Life can indeed be VERY mysterious.
I wanted to remember all this for a long time which is why I spent the hours documenting it above, and in the first 3 articles.
The End
]]>
A Math Mystery and A Math Mystery Follow Up
And after reading this article there’s a 4th and final one
Epilogue – Chronicle of Composites Project
Let’s just start with the formula for the “vanilla” Harmonic Series.
Then massage it in 4 simple steps and see where it takes us!
Note: in the following N is the value of the largest denominator in the “vanilla” Harmonic Series (S_{n} ) and
in the Harmonic Series of Primes (S_{p}). And. of course, assuming N goes to ∞
S_{n} = ln(N)
S_{n} – ln(ln(N)) = ln(N) – ln(ln(N))
S_{n} – S_{p} = ln(N) – ln(ln(N))
S_{n} – S_{p} = ln( N / ln(N) )
S_{n} – S_{p} = ln(π(N))
Now we show the above with comments on the right.
S_{n} = ln(N)

S_{n} is the “vanilla” Harmonic Series 
S_{n} – ln(ln(N)) = ln(N) – ln(ln(N))

let’s subtract ln(ln(N)) from each side of = 
S_{n} – S_{p} = ln(N) – ln(ln(N))

On left side, rewrite ln(ln(N)) as its equivalent which is “Harmonic Series of Primes” (hence the subscript p). 
S_{n} – S_{p} = ln( N / ln(N) )

Doing some log arithmetic, rewrite
ln(N) – ln(ln(N)) as its equivalent of ln( N / ln(N) ) 
S_{n} – S_{p} = ln(π(N))

( N / ln(N) ) is really the “Prime Counting Function” so if we take the ln of it we get ln(π(N)) which is another way of saying that S_{n} – S_{p } over N is the ln of the number of primes over N.
And note that S_{n} – S_{p is, to coin a phrase,} the “Harmonic Series of Composites“ S_{c} = ln(π(N))

So that’s it. In just a few simple steps we are able to show that the “Harmonic Series of Composites” (to coin a phrase) is simply the log of the Prime Counting Function!
S_{c} = ln(π(N))
AND we have seen that the actual computations (thru N = 1 Billion) supports this! But we (I) still do NOT have any intuitive feeling as to WHY!
AND, try as I might, I can NOT find any references to this “discovery” anywhere on the internet. Oh well… time to accept it for what it is… an interesting mystery.
The End
]]>
As They Apply To
Gerry’s Math Mystery Conjecture
Also read prior article A Math Mystery as well as the next 2 articles
Epilogue to the Composites Chronicle
So, this will be the most engrossing blog post ever! Hello!? … Hello!?… Is anybody out there!?
Actually, he meant to say…
You’re gonna need a bigger computer.
This article is a historical record for myself but for some geeky computer types with an interest in math, or vice versa, you may find it of interest. This article has to do with:
1 – The differences in data types and how that affects calculations in our programs; especially programs that calculate things like a numeric series (like the harmonic series… e.g.)
2 – How the Order of calculation also makes a difference in the result. For example, in the above Harmonic Series, does it matter whether we start at the “left”
(1/1 then 1 /2 then 1/3 then… 1/n) or whether we start at the “right” and move back to the beginning of the series
(e.g. 1/n then 1/(n1) then… 1/3 then 1/ 2 then 1/1).
The data types aspect is pretty obvious and is part of Programming 101.
In the prior article, “An Actual Math Mystery,” I used the C# program named SumOfReciprocals.exe to compute a few things over differing ranges. The purpose of the calculations was/is to help confirm, or to dispel the conjecture that:
ln(C_{p}) + S_{p} = S_{n}
Again, read An Actual Math Mystery
The SumOfReciprocals program mainly computes the following two series (plus a few other things). The user specifies the range of values on the screen. Anyway, it computes the following:
“Vanilla Harmonic Series”
Harmonic Series of Primes
The following is an example test run:
The original program used C# doubles as the numeric data type (like so)
double dSumOfVanillaSeries;
double dSumOfPrimeSeries;
With doubles we get 1516 significant digits. But could we do better?
Also, the computation of the two series was performed “left to right.” That is, smallest denominator to highest denominator… 1/1 + 1/ 2 + 1/3 …
As it turns out, the order of evaluation can make a difference. Look here
https://tomroelandts.com/articles/theharmonicseries
So what would the difference/improvement be if we changed the order of evaluation (if any)?
After thinking about these issues for a while, I improved the SumOfReciprocals program by doing the following:
..1.. Variables were changed from double to decimal. This is a big improvement because the decimal data type provides 2829 significant digits vs 1516 for double.
..2.. The order of evaluation was changed to be a user choice. Terms can now be evaluated left–>Right or Right–>Left (user choice via a checkbox).
So… did the improvements to the program make a significant difference? As it turns out, for what I am investigating the improvements in “accuracy” (precision) don’t matter. What I’m after is whether it’s true that
Ln(pi(n)) + S_{p} = S_{n }
which means I use the SumOfReciprocals computations to see whether it looks like
(S_{n} / ( Ln(pi(n)) + S_{p})) ~ 1 as n –> infinity
All that said, I can only run the SumOfReciprocals up to n = 1 Billion which takes about 35 minutes. So, to get results for 1 Trillion would require about 583 hours! I don’t think I’ll be doing that any time soon (although I’d like to).
In any event, and for what it’s worth, the following show a comparison of a few results before and after the program improvements to SumOfReciprocals.exe.
Computing the RATIO (what I’m really interested in)
For n = 1 to 1 Billion
Before improvements using doubles and left–> right computation
RATIO = 1.01251978630066
After improvements using decimals and right–>left computation
RATIO = 1.0125194916706126049689541998
Color shows where the differences begin. Even after improving by going from double to decimal data types and using right–>left processing, for n=1 Billion, the difference in the RATIO is negligible.
So just for grins, let’s see what differences there are with the order of processing the terms… The color coding will help us spot the differences.
n = 1.. 1 Billion
Starting Low and working to higher denominator (left → right)
Harmonic Series Sum = 21.300481502347944016685003449
Harmonic Series of Primes Sum = 3.2927653071574778714154682732
RATIO = 1.0125194916706126049689541998
Starting with higher denominator and working lower (right → left)
Harmonic Series Sum = 21.300481502347944016685100637
Harmonic Series of Primes Sum = 3.2927762759508852619827244781
RATIO = 1.0125189646875907288746445938
The above shows the computation differences for n = 1 .. 1 billion depending on whether we process terms left to right, or, right to left.
So there it stands. Experimentally it still looks like the conjecture is true but again, we’re only computing the two series up to n = 1 billion.
So, was this the most engrossing blog post ever!?
Hello!? … Hello!?… Is anybody out there!?
]]>