some strange numbers

[starsunil]

today i came across a few cool numbers....

they are called antifirst numbers and amicable numbers......

Amicable numbers are a pair of numbers such that the sum of their
proper divisors (not including itself) equals the other number.

Because amicable numbers must come in pairs, they are sometimes called
amicable pairs or amicable partners.

For example, show that 220 and 284 are amicable numbers:

First, find the proper divisors of 220:

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

If you add all these numbers together, you can see they sum to 284.

Now find the proper divisors of 284:

1, 2, 4, 71, 142

These sum to 220.

Thus, 220 and 284 are a pair of amicable numbers.

For another example, let's take the pair 8 and 9:

The proper divisors of 9 are 1 and 3, which sum to 4.
The proper divisors of 8 are 1, 2, and 4, which sum to 7.

So, 8 and 9 are not a pair of amicable numbers since 4 does not equal
8 and 7 does not equal 9.



the anitifirst numbers...

An antifirst number is a number that has more divisors than all of the
numbers before it. For example, the first six antifirst numbers are:

1 It has only one number which divides it (1)
2 It has two (1, 2)
4 It has three (1, 2, 4)
6 It has four (1, 2, 3, 6)
12 It has six (1, 2, 3, 4, 6, 12)
24 It has eight (1, 2, 3, 4, 6, 8, 12, 24)

-------------

dont know how this will be useful......just found it interesting..


sunil

[Debs]

hi all,
contribution to strange number from my side also......... people must b knowing most of them, but still felt that they should b listed together.

3 x 37 = 111 and 1 + 1 + 1 = 3

6 x 37 = 222 and 2 + 2 + 2 = 6
.
.
.
18 x 37 = 666 and 6 + 6 + 6 = 18

21 x 37 = 777 and 7 + 7 + 7 = 21

24 x 37 = 888 and 8 + 8 + 8 = 24

27 x 37 = 999 and 9 + 9 + 9 = 27
--------------------------------------------
1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321
.
.
111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321
----------------------------------------------
1 x 8 + 1 = 9

12 x 8 + 2 = 98
.
.
.
12345 x 8 + 5 = 98765

123456 x 8 + 6 = 987654
.
12345678 x 8 + 8 = 98765432

123456789 x 8 + 9 = 987654321
---------------------------------------
9 x 9 + 7 = 88

98 x 9 + 6 = 888

987 x 9 + 5 = 8888
.
.
987654 x 9 + 2 = 8888888

9876543 x 9 + 1 = 88888888

98765432 x 9 + 0 = 888888888

thats all for today. will get back with some more stuffs. till then, have a nice time.
Bye
Debs

[Debs]

Have someone pick a number between 1 and 9.
Now have him use a calculator to first multiply it by 9, and then
multiply it by 12345679 (notice there is no 8 in that number!).
Have the person show you the result so you can tell him the original
number he selected! How? Easy. If he selected 5, the final answer is
555,555,555. If he selected 3, the final answer is 333,333,333. The
reason: 9 x 12345679 = 111111111. You multiplied your digit by
111111111. (By the way, that 8-digit number (12,345,679) is easily
memorized: only the 8 is missing from the sequence.)

Debs.

[Harry_Potter]

Talking about strange numbers one of them is 142857


If we multiply this number with numbers successively from 1 to 6,

142857 * 1 = 142857
142857 * 2 = 285714
142857 * 3 = 428571
142857 * 4 = 571428
142857 * 5 = 714285
142857 * 6 = 857142

Note that all the products follow a circular pattern.

Also , if we write the products in form of a table and add up the individual coloumns and rows :

1 4 2 8 5 7 | 21
2 8 5 7 1 4 | 21
4 2 8 5 7 1 | 21
5 7 1 4 2 8 | 21
7 1 4 2 8 5 | 21
8 5 7 1 4 2 | 21
-------------
2121212121

If we go a step further ...

142857 * 7 = 999999

Finally , this number is a good way of remembering fractions related with 7 since

1/7 = .142857
2/7 = .285714
3/7 = .428571
4/7 = .571428
5/7 = .714285
6/7 = .857142

[luckynawabs]

If you multiply 1089 x 9 you get 9801. It's reversed itself! This also works with 10989 or 109989 or 1099989 and so on.
1 is the only positive whole number that you can add to 1,000,000 and you get an answer that's bigger than if you multiply it by 1,000,000
19 = 1 x 9 + 1 + 9 and 29 = 2 x 9 + 2 + 9. This also works for 39,49,59,69,79,89 and 99.
153, 370, 371 and 407 are all the "sum of the cubes of their digits". In other words 153=13+53+33
If you divide any square number by 8 you get a remainder of 0, 1 or 4.
2 is the only number that gives the same result added to itself as it does times by itself
If you multiply 21978 by 4 it turns backwards
There are 12,988,816 different ways to cover a chess board with 32 dominoes.
Sixty nine squared = 692 = 4761 and sixty nine cubed = 693 = 328509. These two answers use all the digits from 0 to 9 between them.
You can chop a big lump of cheese into a maximum of 93 bits with 8 straight cuts
In the English language "forty" is the only number that has all its letters in alphabetical order.
1 ÷ 37 = 0·027027027... and 1 ÷ 27 = 0·037037037...
132 = 169 and if you write both numbers backwards you get 312 = 961.
This also works with 12 because 122 = 144 and 212 =441.
1/1089 = 0·00091827364554637281... (And the numbers in the 9 times table are 9,18,27,36...)
8 is the only cube that is 1 less than a square.
To multiply 10,112,359,550,561,797,752,808,988,764,044,943,820 ,224,719 by 9 you just move the 9 at the very end up to the front. It's the only number that does this.

[luckynawabs]

8 = 23 and 9 = 32
These happen to be the only consecutive integers that are powers of primes.
13 & 31 and 17 & 71
These two digit reversals create the following powers of 3:
(31 - 13)/2 = 32
(71 - 17)/2 = 33
24 and 70
sum( n2 ) = 702 where n goes from 1 to 24. This has been proven to be the only sum of squares of consecutive integers (starting at 1), that creates a square.
31 and 8191
31 and 8191 are the only two known numbers (N) that have 3 different cases of the following relation. All the rest have only one or two such cases. Notice that 31 and 8191 are the 3rd and 5th Mersenne primes respectively.
N = (bn - 1)/(b - 1) for 1 < b < N and n > 1
Aside: N is the n-digit number in base b with all digits equal to 1.

31 = (25 - 1) = (53 - 1)/4 = (302 - 1)/29
8191 = (213 - 1) = (903 - 1)/89 = (81902 - 1)/8189

Also N = (3131 - 1)/30 is prime, and so is N = (1919 - 1)/18
55 = 5 x 11
55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
55 = 12 + 22 + 32 + 42 + 52
55 is also the tenth Fibonacci number
89 = the 24th prime
89 interests me since it is the 11th Fibonacci number, and all terms of the Fibonacci sequence are embedded in the decimal expansion of 1/89:
1/89 = 0.01123595506
= 112358
13
21
34
55
89
144
233

Add the Fibonacci numbers, and you get the decimal digits of 1/89. Start
adding from the bottom right, and don't forget to carry. You will need more
terms to obtain more decimal digits.
The prime 2521
(2521-1) x 2 = 7! = 10!/6!
2521 = 352 + 362
2521 x 2 - 1 = 712
2521 - 1 = 2 x 2 x 2 x 3 x 3 x 5 x 7
21 = 3 x 7
And last, but not least, twenty-one just happens to be my favorite number, so I have included it also.
21 is the 8th Fibonacci number. It is also the palindrome 1221 in base 2 of the Alternate Number System and the palindrome 10101 in base 2 of our Existing Number System. Coincidently, I spent my youth at a house which has the number 21, and was born on the 21st of June

[luckynawabs]

8 is largest cube in Fibonacci sequence.
16 is only No. (besides 1) of form xy=yx.
18 is only No. that is twice sum of its digits.
24 is largest No. divisible by all No.s less than its square root.
25 is smallest square that can be written as a sum of 2 squares.
27 is largest No. that is sum of digits of its cube.
2 + 3 + 4 + 5 + 6 + 7 = 27.

[mizzou]

did not get it:
16 is only No. (besides 1) of form xy=yx.

are 17, 23 and many more numbers not same?

[whocarez]

Originally Posted by mizzou @ Mon Sep 13, 2004 6:33 pm

did not get it:
16 is only No. (besides 1) of form xy=yx.

are 17, 23 and many more numbers not same?

he want to say 16 is the only no. which can be expressed in the x^y=y^x (4^2=2^4) where x and y denote 2 different natural numbers....