24.131.80.54: fix link

2014-10-06T13:42:42Z

fix link

← Older revision		Revision as of 15:42, 6 October 2014
Line 1:		Line 1:
	~~In [[mathematics]],~~ a ~~'''highly abundant number''' is a [[natural number]] with~~ the ~~property that the sum of its divisors (including itself)~~ is ~~greater than~~ the ~~sum of~~ the ~~divisors of any smaller natural number~~.		Apologise Nothing beats more than a simple sincere word of apology. When the coast is clear, go inside and hack the computer. When you talk to him end the phone call a couple minutes after you start talking to each other. If you wait for a long period of time she will likely think that you are not interested in her anymore and could start moving on in her own head and heart. And certainly not, the method to bring someone back into our lives following a break up. <br><br>Keep your head up high and don't lose your dignity. The simple truth is that, as humans, we all make mistakes. If he just broke up with you, chances are he really needs his alone time - away from you. " Don't start talking about how much you thought about them during the last few weeks. From that point, I suggest that you look through the upper area for a few things to loot and an ebook. <br><br>Sending a bitter, threatening or abusive text might really send them running very far away from you or even getting a restraining order and in some cases you might even find yourself in jail courtesy of threateniing or abusive charges. Sure you're not officially "dating" anymore, but eventually you'll get her back. It takes a long time to establish a deep connection so it only makes sense that it would take a long time to untangle the connection. Be careful not to analyze things too much, because over analyzing may prevent you from acting the right way when trying to figure out "what can I do to get my ex girlfriend back". Detaching yourself from him gives you an air of self-reliance and by not forcing the issue you make it appear as if you've accepted the separation. <br><br>If you can't convince him, then you'll have to get into the office by hacking the terminal or taking out a guard for some codes. She'll resent you for trying to destroy her new romance, especially since right now she and this new guy are in the honeymoon stages of their relationship. Fourth, once you determine the real reason for the breakup, you will have to do some soul searching. Yes, these three approaches do produce results when it comes to being able to take an ex back. Here's what I did to go from melancholy to success in my love life. <br><br>Whenever your boyfriend breaks up with you but asks to remain friends, it's both a blessing and a curse. Whatever your motivation is to get your ex back, you may be clueless as to how to even start this process and do it in a way that you won't seem too desperate. Actually, anything you say or even do during this period and regardless of what your objectives are, can only exacerbate the situation and perhaps drive your ex wife farther away from you. You want him back because he dumped you and made you feel rejected and unwanted. Try to understand that your boyfriend ignoring you means absolutely nothing right now.<br><br>If you cherished this informative article and also you desire to obtain more info regarding get him back forever ([http://www.kok1235.info/ http://www.kok1235.info/]) generously pay a visit to the web-site.

	~~Highly abundant numbers and several similar classes~~ of ~~numbers were first introduced by [[Subbayya Sivasankaranarayana Pillai\|Pillai]] (1943),~~ and ~~early work~~ on ~~the subject was done by [[Leonidas Alaoglu\|Alaoglu]]~~ and ~~[[Paul Erdős\|Erdős]] (1944)~~. ~~Alaoglu and Erdős tabulated all highly abundant numbers up to 10<sup>4</sup>~~, ~~and showed that~~ the ~~number of highly abundant numbers less than any ''N'' is at least proportional~~ to ~~log~~<~~sup~~>2<~~/sup~~> '~~'N''~~. ~~They also proved~~ that ~~7200 is the largest [[powerful number\|powerful]] highly abundant number~~, ~~and therefore~~ the ~~largest highly abundant number with odd sum of divisors~~.

	~~== Formal definition and examples ==~~

	~~Formally~~, a ~~natural number ''n'' is called highly abundant [[if~~ and ~~only if]] for all natural numbers ''m'' < ''n'',~~

	:<~~math>\sigma(n)~~ > ~~\sigma(m)~~<~~/math~~>

	~~where σ denotes the [[divisor function\|sum-~~of~~-divisors function]]~~. ~~The first few highly abundant numbers are~~
	:[[1 (number)\|1]], [[2 (number)\|2]], [[3 (number)\|3]], [[4 (number)\|4]], [[6 (number)\|6]], [[8 (number)\|8]], [[10 (number)\|10]], [[12 (number)\|12]], [[16 (number)\|16]], [[18 (number)\|18]], [[20 (number)\|20]], [[24 (number)\|24]], [[30 (number)\|30]], [[36 (number)\|36]], [[42 (number)\|42]], [[48 (number)\|48]], [[60 (number)\|60]], ..~~. {{OEIS\|id=A002093}}.~~

	~~For instance, 5 is~~ not ~~highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7~~, ~~while 8 is highly abundant~~ because ~~σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ~~.

	~~== Relations with other sets~~ of ~~numbers ==~~

	~~Although~~ the ~~first eight factorials are highly abundant~~, ~~not all factorials are highly abundant~~. ~~For example~~,
	~~:σ(9!) = σ(362880) = 1481040,~~
	~~but there is a smaller number with larger sum~~ of ~~divisors~~,
	~~:σ(360360) = 1572480~~,
	~~so 9! is not highly abundant~~.

	~~Alaoglu and Erdős noted that all [[superabundant number]]s are highly abundant~~, ~~and asked whether there are infinitely many highly abundant numbers that are not superabundant~~. ~~This question was answered affirmatively by Nicolas (1969)~~.

	~~Despite the terminology~~, ~~not all highly abundant numbers are [[abundant number]]~~s. ~~In particular, none of the first seven highly abundant numbers~~ is ~~abundant.~~

	~~== References ==~~

	*{{cite journal
	~~\| author = [[Leonidas Alaoglu\|Alaoglu~~, L.~~]]; [[Paul Erdős\|Erdős~~, ~~P.]]~~
	~~\| title = On highly composite~~ and ~~similar numbers~~
	~~\| journal = [[Transactions~~ of the ~~American Mathematical Society]]~~
	~~\| volume = 56~~
	~~\| year = 1944~~
	~~\| pages = 448–469~~
	~~\| mr = 0011087~~
	~~\| doi = 10~~.~~2307/1990319~~
	~~\| issue = 3~~
	~~\| jstor = 1990319}}~~

	*{{cite journal
	~~\| author = Nicolas, Jean-Louis~~
	~~\| title = Ordre maximal d'un élément du groupe ''S~~<~~sub~~>n</~~sub>'' des permutations et "highly composite numbers"~~
	~~\| journal = Bull~~. ~~Soc~~. ~~Math. France~~
	~~\| volume = 97~~
	~~\| year = 1969~~
	~~\| pages = 129–191~~
	~~\| mr = 0254130~~
	~~\| url =~~ http://www.~~numdam~~.~~org~~/~~item?id=BSMF_1969__97__129_0}}~~

	*{{cite journal
	~~\| author = Pillai, S. S.~~
	~~\| authorlink = Subbayya Sivasankaranarayana Pillai~~
	~~\| title = Highly abundant numbers~~
	~~\| journal = Bull~~. ~~Calcutta Math. Soc.~~
	~~\| volume = 35~~
	~~\| year = 1943~~
	~~\| pages = 141–156~~
	~~\| mr = 0010560 }}~~


	~~{{Divisor classes}}~~
	~~{{Classes of natural numbers}}~~

	~~[[Category:Divisor function]]~~
	~~[[Category:Integer sequences]]~~

en>EtymAesthete at 22:11, 4 September 2013

2013-09-04T22:11:40Z

Show changes

en>MindZiper at 03:35, 9 November 2011

2011-11-09T03:35:54Z

Show changes

← Older revision		Revision as of 15:42, 6 October 2014
Line 1:		Line 1:
	~~In [[mathematics]],~~ a ~~'''highly abundant number''' is a [[natural number]] with~~ the ~~property that the sum of its divisors (including itself)~~ is ~~greater than~~ the ~~sum of~~ the ~~divisors of any smaller natural number~~.		Apologise Nothing beats more than a simple sincere word of apology. When the coast is clear, go inside and hack the computer. When you talk to him end the phone call a couple minutes after you start talking to each other. If you wait for a long period of time she will likely think that you are not interested in her anymore and could start moving on in her own head and heart. And certainly not, the method to bring someone back into our lives following a break up. <br><br>Keep your head up high and don't lose your dignity. The simple truth is that, as humans, we all make mistakes. If he just broke up with you, chances are he really needs his alone time - away from you. " Don't start talking about how much you thought about them during the last few weeks. From that point, I suggest that you look through the upper area for a few things to loot and an ebook. <br><br>Sending a bitter, threatening or abusive text might really send them running very far away from you or even getting a restraining order and in some cases you might even find yourself in jail courtesy of threateniing or abusive charges. Sure you're not officially "dating" anymore, but eventually you'll get her back. It takes a long time to establish a deep connection so it only makes sense that it would take a long time to untangle the connection. Be careful not to analyze things too much, because over analyzing may prevent you from acting the right way when trying to figure out "what can I do to get my ex girlfriend back". Detaching yourself from him gives you an air of self-reliance and by not forcing the issue you make it appear as if you've accepted the separation. <br><br>If you can't convince him, then you'll have to get into the office by hacking the terminal or taking out a guard for some codes. She'll resent you for trying to destroy her new romance, especially since right now she and this new guy are in the honeymoon stages of their relationship. Fourth, once you determine the real reason for the breakup, you will have to do some soul searching. Yes, these three approaches do produce results when it comes to being able to take an ex back. Here's what I did to go from melancholy to success in my love life. <br><br>Whenever your boyfriend breaks up with you but asks to remain friends, it's both a blessing and a curse. Whatever your motivation is to get your ex back, you may be clueless as to how to even start this process and do it in a way that you won't seem too desperate. Actually, anything you say or even do during this period and regardless of what your objectives are, can only exacerbate the situation and perhaps drive your ex wife farther away from you. You want him back because he dumped you and made you feel rejected and unwanted. Try to understand that your boyfriend ignoring you means absolutely nothing right now.<br><br>If you cherished this informative article and also you desire to obtain more info regarding get him back forever ([http://www.kok1235.info/ http://www.kok1235.info/]) generously pay a visit to the web-site.

	~~Highly abundant numbers and several similar classes~~ of ~~numbers were first introduced by [[Subbayya Sivasankaranarayana Pillai\|Pillai]] (1943),~~ and ~~early work~~ on ~~the subject was done by [[Leonidas Alaoglu\|Alaoglu]]~~ and ~~[[Paul Erdős\|Erdős]] (1944)~~. ~~Alaoglu and Erdős tabulated all highly abundant numbers up to 10<sup>4</sup>~~, ~~and showed that~~ the ~~number of highly abundant numbers less than any ''N'' is at least proportional~~ to ~~log~~<~~sup~~>2<~~/sup~~> '~~'N''~~. ~~They also proved~~ that ~~7200 is the largest [[powerful number\|powerful]] highly abundant number~~, ~~and therefore~~ the ~~largest highly abundant number with odd sum of divisors~~.

	~~== Formal definition and examples ==~~

	~~Formally~~, a ~~natural number ''n'' is called highly abundant [[if~~ and ~~only if]] for all natural numbers ''m'' < ''n'',~~

	:<~~math>\sigma(n)~~ > ~~\sigma(m)~~<~~/math~~>

	~~where σ denotes the [[divisor function\|sum-~~of~~-divisors function]]~~. ~~The first few highly abundant numbers are~~
	:[[1 (number)\|1]], [[2 (number)\|2]], [[3 (number)\|3]], [[4 (number)\|4]], [[6 (number)\|6]], [[8 (number)\|8]], [[10 (number)\|10]], [[12 (number)\|12]], [[16 (number)\|16]], [[18 (number)\|18]], [[20 (number)\|20]], [[24 (number)\|24]], [[30 (number)\|30]], [[36 (number)\|36]], [[42 (number)\|42]], [[48 (number)\|48]], [[60 (number)\|60]], ..~~. {{OEIS\|id=A002093}}.~~

	~~For instance, 5 is~~ not ~~highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7~~, ~~while 8 is highly abundant~~ because ~~σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ~~.

	~~== Relations with other sets~~ of ~~numbers ==~~

	~~Although~~ the ~~first eight factorials are highly abundant~~, ~~not all factorials are highly abundant~~. ~~For example~~,
	~~:σ(9!) = σ(362880) = 1481040,~~
	~~but there is a smaller number with larger sum~~ of ~~divisors~~,
	~~:σ(360360) = 1572480~~,
	~~so 9! is not highly abundant~~.

	~~Alaoglu and Erdős noted that all [[superabundant number]]s are highly abundant~~, ~~and asked whether there are infinitely many highly abundant numbers that are not superabundant~~. ~~This question was answered affirmatively by Nicolas (1969)~~.

	~~Despite the terminology~~, ~~not all highly abundant numbers are [[abundant number]]~~s. ~~In particular, none of the first seven highly abundant numbers~~ is ~~abundant.~~

	~~== References ==~~

	*{{cite journal
	~~\| author = [[Leonidas Alaoglu\|Alaoglu~~, L.~~]]; [[Paul Erdős\|Erdős~~, ~~P.]]~~
	~~\| title = On highly composite~~ and ~~similar numbers~~
	~~\| journal = [[Transactions~~ of the ~~American Mathematical Society]]~~
	~~\| volume = 56~~
	~~\| year = 1944~~
	~~\| pages = 448–469~~
	~~\| mr = 0011087~~
	~~\| doi = 10~~.~~2307/1990319~~
	~~\| issue = 3~~
	~~\| jstor = 1990319}}~~

	*{{cite journal
	~~\| author = Nicolas, Jean-Louis~~
	~~\| title = Ordre maximal d'un élément du groupe ''S~~<~~sub~~>n</~~sub>'' des permutations et "highly composite numbers"~~
	~~\| journal = Bull~~. ~~Soc~~. ~~Math. France~~
	~~\| volume = 97~~
	~~\| year = 1969~~
	~~\| pages = 129–191~~
	~~\| mr = 0254130~~
	~~\| url =~~ http://www.~~numdam~~.~~org~~/~~item?id=BSMF_1969__97__129_0}}~~

	*{{cite journal
	~~\| author = Pillai, S. S.~~
	~~\| authorlink = Subbayya Sivasankaranarayana Pillai~~
	~~\| title = Highly abundant numbers~~
	~~\| journal = Bull~~. ~~Calcutta Math. Soc.~~
	~~\| volume = 35~~
	~~\| year = 1943~~
	~~\| pages = 141–156~~
	~~\| mr = 0010560 }}~~


	~~{{Divisor classes}}~~
	~~{{Classes of natural numbers}}~~

	~~[[Category:Divisor function]]~~
	~~[[Category:Integer sequences]]~~

Propargyl - Revision history

24.131.80.54: fix link

en>EtymAesthete at 22:11, 4 September 2013

en>MindZiper at 03:35, 9 November 2011