Strange Numbers

[Turtle]

Originally Posted by TheBigDog

I check these numbers. The second is indeed Curious, but has perfect number 28 as a factor, so it is not anomalous, or Exceptional. The first is also curious, but has perfect number 6 as a factor. I am adding a tool to the Numberator for entering and testing any number of your choice to aid in this type of research!

Bill

Whoa! Again. This is a new discovery & merits closer scrutiny. Remebering that the standard run-o'-the-mill Curious Number has 14 pairs of divisors, so my two 'exceptions' differ in this regard. I feel confident to say it's an instance of 'Exceptional Anomalous Curious Numbers'.
Moreover, the run-o'-the'mill Curious Number is abundant by twice the Perfect 8128, not 6 or 28. This is more-or-less the set definition as I laid it out.
Speculatous interuptous; someone came in to talk to me & I lost my train ....

[Turtle]

Just looking over my lists & I spotted 9424. It is the ninth Peculiar Number & it has the first exceptional Strange Number paired with a prime as factors (304*31)
I seem to have misplaced my list of exceptional Peculiars.

[TheBigDog]

Originally Posted by Turtle

Whoa! Again. This is a new discovery & merits closer scrutiny. Remebering that the standard run-o'-the-mill Curious Number has 14 pairs of divisors, so my two 'exceptions' differ in this regard. I feel confident to say it's an instance of 'Exceptional Anomalous Curious Numbers'.
Moreover, the run-o'-the'mill Curious Number is abundant by twice the Perfect 8128, not 6 or 28. This is more-or-less the set definition as I laid it out.
Speculatous interuptous; someone came in to talk to me & I lost my train ....

Hmmm.... Just a bump in the road. That is why I keep my knees bent while doing math at high speed.

Here is the process that the program is doing.

1. See if the number qualifies for the "strange" family by comparing the sum of factors to the number
2. If the number is perfect add to the list of perfects and move to the next number
3. If not in the strang family throw it away and move to next number
4. If in the "strange" family check to see if any of the factors are perfect
5. If any of the factors are perfect then list as a normal strange number
6. If none of the factors are perfect then list as an anomalous strange number

I take it from your notes that I may have an error. Should I only be checking numbers against their own root perfect number, so to speak?

Bill

[TheBigDog]

Originally Posted by Turtle

Just looking over my lists & I spotted 9424. It is the ninth Peculiar Number & it has the first exceptional Strange Number paired with a prime as factors (304*31)
I seem to have misplaced my list of exceptional Peculiars.

I am going to put the factors into a list that shows info about each one. What are some of the qualities of the factors that you want at your fingertips?

Bill

[Turtle]

Originally Posted by TheBigDog

Here is the process that the program is doing.

1. See if the number qualifies for the "strange" family by comparing the sum of factors to the number
2. If the number is perfect add to the list of perfects and move to the next number
3. If not in the strang family throw it away and move to next number
4. If in the "strange" family check to see if any of the factors are perfect
5. If any of the factors are perfect then list as a normal strange number
6. If none of the factors are perfect then list as an anomalous strange number

I take it from your notes that I may have an error. Should I only be checking numbers against their own root perfect number, so to speak?

Bill

Yes, that sounds like the solution. I set my test for Strange for example by looking for abundance by 12. Early on when I first found the sets I did not even realize the 'abundance amounts' were double the Perfect Numbers. That's serendipity.
The main exception(s) then have no 'root Perfect' factors & a different number of factor pairs than standard issue.

[TheBigDog]

Originally Posted by Turtle

Yes, that sounds like the solution. I set my test for Strange for example by looking for abundance by 12. Early on when I first found the sets I did not even realize the 'abundance amounts' were double the Perfect Numbers. That's serendipity.
The main exception(s) then have no 'root Perfect' factors & a different number of factor pairs than standard issue.

That may make it even faster!

Bill

[Turtle]

Originally Posted by TheBigDog

I am going to put the factors into a list that shows info about each one. What are some of the qualities of the factors that you want at your fingertips?

Bill

I (we) don't know yet; that's the fun of it. I'd say that having a formatted extensive list is a good start. Keeping Strange Set, Bizarre Set, etc. separate, listing the exception & its factors/factor pairs. While we do not count the number itself in the sum, I think it belongs on the list paired with 1 for clarity.
Where's Waldo now?

[TheBigDog]

OK, I have my work before me. I will give an update soon. This will be Numberator 2.0 - Rise of the Katabatak! (with definable base)

Public Function k(ByVal num as Long, Optional ByVal base as Integer = 10)

Bill

[Turtle]

Originally Posted by TheBigDog

OK, I have my work before me. I will give an update soon.
Bill

No pressure. Count your lucky stars that I don't go after this math like Paul Erdos went after his. In such a case I would simply show up at your house uinvited & stay until we proved something significant of all this.

[Turtle]

If you find yourself asking 'what's going on here?' with all these funny names of numbers & the whole kit-n'-kaboodle of this thread, then you have gathered in good company. Here's a summary to bide our time while TheBigDog works behind the scenes.
We all know what it means to say a number divides evenly, and more so we know how to find the different ways a number divides evenly. We say the numbers that evenly divide another number are the factors (or divisors). Nothing like an example & I choose 8. What numbers evenly divide it? 1, of course, and 2, and 4, and then 8 itself because every number evenly divides itself.
The ancient Greeks then took the step of adding together all the factors except the number itself, and then comparing the resulting sum to the number they came from. For the example we add 1 + 2 + 4 = 7. This comparison has three possible results; less than, greater than, or equal to.
The Greeks named these three divisions 'Deficient' if the sum is less, 'Perfect' if the sum is equal, and 'Abundant' if the sum is greater than.
Once you have this method of classifying numbers, you of course set about to systematically classify them all in order. Enter the programmable computer & bigger numbers then ever before classified.
Back to our Strange & other Unusual sets of number then. What they are is a special subset of the Abundant Numbers; special by virtue of 'how much' they are Abundant by. A number Abundant by 12, is Strange, simply by definition just as a number that divides by just 1 & itself is Prime.
We now return to the regular station.