F-distribution: Difference between revisions

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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you loved this article and also you would like to collect more info about [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] nicely visit our web site.
'''Sociable numbers''' are numbers whose [[Aliquot_sum#Definition|aliquot sums]] form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of [[amicable number]]s and [[perfect number]]s. The first two sociable sequences, or sociable chains, were discovered and named by the [[Belgium|Belgian]] [[mathematics|mathematician]] [[Paul Poulet (mathematician)|Paul Poulet]] in 1918. In a set of sociable numbers, each number is the sum of the [[proper factors]] of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.
 
The [[Frequency|period]] of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
 
If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the [[proper divisor]]s of 6 are 1, 2, and 3, whose sum is again 6. A pair of [[amicable number]]s is a set of sociable numbers of order 2. There are no known sociable numbers of order 3.
 
It is an open question whether all numbers end up at either a sociable number or at a [[Prime number|prime]] (and hence 1), or, equivalently, whether there exist numbers whose [[aliquot sequence]] never terminates, and hence grows without bound.
 
An example with period 4:
:The sum of the proper divisors of <math>1264460</math> (<math>=2^2\cdot5\cdot17\cdot3719</math>) is:
::1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860
 
:The sum of the proper divisors of <math>1547860</math> (<math>=2^2\cdot5\cdot193\cdot401</math>) is:
::1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636
 
:The sum of the proper divisors of <math>1727636</math> (<math>=2^2\cdot521\cdot829</math>) is:
::1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
 
:The sum of the proper divisors of <math>1305184</math> (<math>=2^5\cdot40787</math>) is:
::1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
 
The following categorizes all known sociable numbers as of October 2009 by the length of the corresponding aliquot sequence:
 
{| align="center" border="1" cellpadding="4"
|- bgcolor="#A0E0A0" align="center"
!Sequence
length
!Number of  
sequences
|- align="center"
| 1
(''perfect'')
| 47
|- align="center"
| 2
(''amicable'')
|11,994,387
|- align="center"
|4
|165
|-align="center"
|5
|1
|- align="center"
|6
|5
|- align="center"
|8
|2
|- align="center"
|9
|1
|- align="center"
|28
|1
|}
 
==References==
*P. Poulet, #4865, [[L'Intermédiaire des Mathématiciens]] '''25''' (1918), pp. 100-101.
*H. Cohen, ''On amicable and sociable numbers,'' Math. Comp. '''24''' (1970), pp. 423-429
 
== External links ==
*[http://djm.cc/sociable.txt A list of known sociable numbers]
*[http://amicable.homepage.dk/tables.htm Extensive tables of perfect, amicable and sociable numbers]
*{{mathworld |urlname=SociableNumbers |title=Sociable numbers}}
 
 
{{Divisor classes}}
{{Classes of natural numbers}}
 
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Number theory]]

Revision as of 15:34, 1 February 2014

Sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a set of sociable numbers, each number is the sum of the proper factors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.

The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.

If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3.

It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.

An example with period 4:

The sum of the proper divisors of 1264460 (=22⋅5⋅17⋅3719) is:
1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860
The sum of the proper divisors of 1547860 (=22⋅5⋅193⋅401) is:
1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636
The sum of the proper divisors of 1727636 (=22⋅521⋅829) is:
1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
The sum of the proper divisors of 1305184 (=25⋅40787) is:
1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.

The following categorizes all known sociable numbers as of October 2009 by the length of the corresponding aliquot sequence:

Sequence

length

Number of

sequences

1

(perfect)

47
2

(amicable)

11,994,387
4 165
5 1
6 5
8 2
9 1
28 1

References

External links


Template:Divisor classes Template:Classes of natural numbers

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