Strange Numbers

[alxian]

with integers, and you are telling me that larger numbers have small numbers of such integer divisors?

 

cool.. still following you.

 

whats the largest number you've processed with the smallest number if interger divisors?

[Qfwfq]

To be Strange the difference must be 12. You can find most Strange numbers by multiplying 6 (a perfect number) times a prime. But this does not find 304, which does not divide by 6(or 12). Also most Strange numbers have exactly 8 divisors (4 pairs) & 304 has 10 (5 pairs) This why it's strange. 304 I call an anomalous Strange Number & the next one is 127,744.

It is easy to show why 6p is strange for any prime p and also why it has 8 divisors. There is no reason for the opposite implications to hold, hence cases such as 304 are free to occur. As usual, it is easier to use the sum including the number itself so that perfect means S = 2n and strange means S = 2n + 12. Multiply 6 by any prime p and its divisors wil be:

 

1, 2, 3, 6, p, 2p, 3p, 6p

 

no more, no less than 8 of them. The sum is exactly 12 + 12p = 12 + 2n.

[Tormod]

Sounds like someone is making progress here. :naughty:

[eddyozzz]

Hi everyone, I was reading about the strange numbers and your great commands, one thing though. According to the definition I have; An odd positive integer greater then 3 is called a strange number if it can not be expressed as the sum of a positive power of 2 and a prime number. 24,30,42,54,66,78,102 all seem to be strange but they are not odd though...please correct me if i am wrong guys. I just have to find the first five strange numbers.

[eddyozzz]

Hello everyone, I am a new member :) , I am trying to find the first 6 strange numbers, in the definition i have it says that an odd positive integer greater than 3 called strange number if it cant be expressed as the sum of a positive power of a 2 and a prime number.

[Qfwfq]

Welcome Eddy.

 

The definition of strange used here is one chosen by Turtle. The name he chose might already be in use.

 

Note that the definition you report couldn't give an odd number as it will give multiples of 2, unless you include the power 2 raised to zero. Perhaps the definition you saw subtracts 1 at the end:

 

p*2^n - 1

 

In this case, they would be a dime a dozen.

[eddyozzz]

Thanks for the reply folks. As as example ; lets look at 9(which is an odd). We can write 2^3+1 = 9 since 1 isnt a prime it seems ok but ,9 can also be expressed 2^2+5 where 5 is a prime so since we have 1 set of expression with a prime ,that means 9 is not a strange number. Another example; 17 ; 2^3+9 is ok since 9 isnt a prime, 2^1+15 is ok, but 2^2+13 opps 13 is a prime so 17 can not be a strange number. I use mathemetica to come up with an answer to this becase thats where i got the definition to the strange number but i just can not put the codes together to come up with an strange numbers. It must be someting has to do with gcd(x,y)...

[Qfwfq]

expressed as the sum of a positive power of a 2 and a prime number.

Very silly, I hadn't read properly. Sum you say, not product.

 

Well that's a different kettle of fish! As you get to higher odd numbers there will be more and more powers of two you can subtract, in each case the difference will be odd, and so there is an increasing likelihood of it not being strange according to that definition. If you take a number greater than 2 raised to the 999th power it'll be an awful long job to check for sure that it is strange.

 

Greatest common divisor... I don't see how it would help much but I'll think the matter over through Easter.

 

Note: you asked about the even numbers that "appear" strange. For an even number minus a power of 2 the difference will clearly be even, unless this difference is 2 it can't be a prime. This must be why that definition was chosen specifying odd. Even and apparently strange numbers really would be a dime a dozen.

[Queso]

this stuff intrigues me to the bone, but i hardly understand it. the things i grasp are only bricks in this giant wall of your mind.

[Queso]

.

___I consider this topic in high regard as it began with an epiphany & continues as a mystery. ;)

ah, just like life!

[TheBigDog]

Whew! I just read this whole thread from beginning to end. I am very impressed indeed Turtle. I am down right intreagued. I had never heard of strange numbers before. Or thought of the relationship of the sum of the factors of a number to the quality of the number itself (in fact I would have probalby just added up the prime factors as I am a prime nut). Triangles, as you know, have taken some brain time from me. And I have long been fond of prime numbers and good old Fibonocci. But this is like a new flavor of candy and I can't wait to taste it. I am going to write a program for searching out these numbers. I would be interested in the TurboBasic program you used for your previous searches to give me a starting point. I will keep you advised as to my progress.

 

This is going to be fun!

 

Bill

[TheBigDog]

___Ok, you shall have it. It so happens I converted one version to a text file today; I have attached it below. I ran it to see what it did & it appears to work, but it is a modified version of what I made the discovery with. Then there is the matter of...hedging.:hyper: I know a lot of people will chuckle at my style, but I throw caution to the wind & here it is unedited. So ...uh... here it is.:hyper:

Very cool! I am going to be working in vb.net. I will send the text of my code when I am done if you are interested. I want to make this one really jazzy.

 

Bill

[TheBigDog]

I have been consumed by the Fraculator. I have been working on version 2 which includes many new functions. It is letting me brush up on my programming which had been sitting idle for about 12 months. Strange numbers is next on the list. I am going to take the code from you and try and build some interactive analytical tools around it. I will get the working tool to you in some fashion or another.

 

:smart: :evil: ;)

 

Now go spot Sasquatch! :eek_big:

 

Bill

[Racoon]

Right On BigDog! :eek_big: :cup: .

 

Turtle filled my head with some "Strange" ideas too! :xx:

Cardinal numbers of Unusual sets??

Does that ring a bell?

 

I still can't tell if 16 is abundant or deficient...

1 x 16

8 x 2

*4 x 4* <<<<:smart: :evil:

 

I hope you save some time for work and the kids.;)

We need more RFID tags :)

[TheBigDog]

Howdy Racoon!

 

I am so exhausted. I do this to relax.

 

Bryan! Stop running!

 

#4 did a faceplant off of a desk at his daycare. :evil: His face is swollen so he looks like he should be on the Simpsons. Knocked his front two teeth loose. I get to take him to the dentist on Monday (during work some time) to find out if they have a chance of staying in. Taking #2, #3 and #4 to a hockey game tomorrow night. #2 is a hockey nut. These tickets are his birthday present. He has been waiting since Dec 31 for this game. #4 will fit right in. The boys are really excited. We are going to paint our faces for the game. We may do some body painting too. And #D is sleeping across my feet - he knows who the BigDog is. Shannon was in the hospital last weekend because the doctors thought she had apendicitis. She is fine now, but I am still catching up on sleep from her being out of it. I am aiming to have the color selector done tonight on the Fractalator. Might have it posted by Sunday. I still have to watch the season ender of Battlestar Galactica. And I am way behind on somme of my favorite threads.

 

Bryan! Stop running!

 

So you can see, I do this to relax. I am looking forward to some quality time with strange numbers. :eek_big:

 

Bill

[TheBigDog]

Rumor has it that some new software is in the making to expand the exploration of these 'Unusual' sets of numbers.

:wave:

:circle:

My anticipation is palpable!

During the periods I obsessed on these sets, I seldom stopped the program to code 'extra' features. Well, worse than that even, but I just wanted to plunge ahead to the real discovery stuff, not polish my programming. The one thing that I didn't code for is writing the output to files, & that has made for problems ever since. I either watched the screen for hours for numbers to appear that I wrote traps for, or set a certain range & printed output to a line printer.

I do run on; sorry. I'll just go off now & look for some tight socks.:phones:

It has begun.... Be prepared for questions.

 

:) :xx: :smart: :edizzy: :hihi: :rant: :hihi: :tongue: :hihi: :singer: :singer:

 

Stand back numbers! I mean business!

 

Bill

[TheBigDog]

Question #1 - Do we want to keep he results for all numbers, or just the interesting numbers?

 

Bill