Strange Numbers

[Turtle]

It is not that the difference "needs" to be 12, & yes, it can be ( and is) almost everything else. The simple point is, all those other differences that "be", don't form up any consistant sets. Differnces of twice a perfect do. I don't know why, I only know the property exists. Keep at me until you understand.

[Turtle]

Here's a little 'splainin:

My computer experiment in factoring by which I found the Strange Numbers & others in their class is rather simple. In sequence, take an integer & factor it, sum its divisors (except don't count the number itself ala perfect number definition), then subtract the number from its sum.
Now display on the screen these three values one above the other; the integer, its sum of divisors & the difference. Then start the loop to sequentially factor the integers & set back & watch the display. Mind you I started this all on an 8088 so it went slow enough to let me see the patterns.
Now I color coded the difference display so it was yellow if the sum<integer (ie. deficient) & purple if the sum >integer. What I noticed was a predominately random appearing set of values in the difference, except for some purple (abundant) differnces that seemed regular. Regularlly appearing I saw 12, 56,& 992.
I stopped the run & wrote a trap to seperately display any integers giving this difference of 12, 56, or 992. To my delight I found these sets had mostly members with the same numbers of divisor pairs ( 4 for Strange). Further, most the set's elements had a pair of divisors that is a perfect & a prime. And still further most set elements differ from one another by an even factor of a perfect number. More intriqung yet are the anamolies such as 304 in the Strange set; it belongs because the sum of its divisors excedes it by 12,(the very definition of set membership) yet it has 5 divisor pairs, no perfect factors, & not differnce from its fellow set members that is an even multiple of 6.
Now I have done runs from 1 to over 13 million & lasting weeks as well as runs beginning in the 100 millions & watch the difference display as you like, the sets I found have no corallary in differnce values other than if the differnce is twice a perfect.

[maddog]

Originally Posted by Turtle

It is not that the difference "needs" to be 12, & yes, it can be ( and is) almost everything else. The simple point is, all those other differences that "be", don't form up any consistant sets. Differnces of twice a perfect do. I don't know why, I only know the property exists. Keep at me until you understand.

I read your explanation and I now getting it. I am wondering if we are seeing hear something
like a number theory version of fractals. I see playing with the two operations of addition and
multiplication over the integers, thus we are dealing with ring. So I am wondering if these
form some prime ideals. I am sorry if I am sound gibberish; these are terms from Group
Theory and Abstract Algebra. I will ask a friend of mine who is a Number Theorist here and
see what he thinks. I am intrigued.

What if I were to construct this over the complex integers {x + iy} and do the same amount
of computation over its modulus ? I probably won't get a chance to discuss with him until
next week. Good Luck!

[Turtle]

No sorry needed; this is similar to my approach to these experiments. I draw nomenclature, functions, etc from just such diverse areas of math as you mention & use them as I would tools from my box. I could plane an inch off a board, but why not saw it 7/8 & just plane 1/8? You seem on the right track & thankyou for offering to share this.

[Turtle]

The first few Strange Numbers:
The Infinite Set S, Strange Numbers - Base Ten
{24, 30,42, 54, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, ...}

The first few Peculiar Numbers

The First Ten Elements Of Set P (Peculiar numbers)
{1488, 2480, 2892, 3472, 5456, 6104, 6448, 8432, 9424, 11408, ...}

[maddog]

Turtle,

I talked with my Number Theorist friend and he said he had played with some of this earlier
in his development. He showed me a generalization by picking another example based on
28 (another Perfect number) as follows:

Let p be a prime number, q be a perfect number, g and k be an integer. Any number of the
form

g = q * p^k

will also have Strange properties as you mention. They need not nessecarily be always 12
more (but any perfect number)

This does increase the overall number of Strange numbers and change their distribution.

A corrolary I am now wondering, are all Perfect numbers divisible by 2 ?

Maddog

[Turtle]

Now we're getting somewhere. Your friend's general expression is not enough & too much. You don't need to raise the prime to any power at all & it will not produce all set memebers. 304 is a Strange Number & has no perfect factors. Now I call the set Strange when the divisor sum is 12 more, Bizarre set numbers divisor sum 56 more, peculiar 992 more, Curious set is 16,256 more; yes each excedes by 2*perfect. Of course the question of proving if all perfect numbers are even is an unsolved problem in number theory; a kind of holy grail still. Now I don't accept that your friend is at all describing what I found & still claim I am the first to describe these sets. Search as you will, you will find no list elewhere which arranges these numbers in a set as I have, nor any expression that produces them.

[Turtle]

Another interesting thing about these sets; except for the second element of each set, and except for elements of the sets that have no perfect factors, no elements of these sets have 5 as a factor.

[maddog]

Originally Posted by Turtle

Now we're getting somewhere. Your friend's general expression is not enough & too much. You don't need to raise the prime to any power at all & it will not produce all set memebers.

My memory is not too good. I must have added the powers. BTW, he was calling them
Wierd numbers with the same definition as you describe.

Originally Posted by Turtle

304 is a Strange Number & has no perfect factors. Now I call the set Strange when the divisor sum is 12 more, Bizarre set numbers divisor sum 56 more, peculiar 992 more, Curious set is 16,256 more; yes each excedes by 2*perfect. Of course the question of proving if all perfect numbers are even is an unsolved problem in number theory; a kind of holy grail still. Now I don't accept that your friend is at all describing what I found & still claim I am the first to describe these sets. Search as you will, you will find no list elewhere which arranges these numbers in a set as I have, nor any expression that produces them.

Turtle - how is 304 Strange when it is the product of 2^4 * 19 = 304. So the factors:

304: 2, 4, 8, 16, 19, 38, 76, 152 => sum = 315 which is 11 more than 304 and Not 12 ?

I don't know if he found what specifically you found (?), what my friend said is a way to
represent "all" the numbers as

g = q * p where q is a perfect integer and p is a prime (sorry for the powers earlier).

Maddog

[Turtle]

You forgot to include 1 in summing the divisors; remember again a perfect number is one whose sum of its proper divisors (including 1, but not the number itself) is equal to it. 6 is considred the first perfect, although a logical argument for 1 as perfect is simple. So, you were off by one; a common pitfall in number theory. So, 304 is Strange. (Oh; weird numbers deal with sums of divisors, but not as Strange Numbers do).
Between 1 & 12,000,000 are a bunch of Strange Numbers, but only 2 aren't of the form prime * perfect. Those 2 are 304 & 127,744. (As far as I can tell from my notes.) You make me see maybe we only consider these anamolous members of the set. Now, you are in check again Sir King Dog; you must factor( considered a hard computer task;thats why most encryption is based on large primes or pseudo primes) to find all members of the set. Thanks for stringin this thread along.
PS Were you just testing me with that powers thing?