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The '''spt function''' (smallest parts function) is a function in [[number theory]] that counts the sum of the number of smallest parts in each [[Partition (number theory)|partition]] of a positive integer. It is related to the [[Partition function (number theory)|partition function]].
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The first few values of spt(''n'') are:
 
:1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589 ... {{OEIS|id=A092269}}
 
== Example ==
 
For example, there are five partitions of 4: (1,1,1,1), (1,1,2), (1,3), (2,2) and (4). These partitions have 4, 2, 1, 2 and 1 smallest parts respectively. So spt(4) = 4 + 2 + 1 + 2 + 1 = 10.
 
== Properties ==
 
Like the partition function, spt(''n'') has a [[generating function]]. It is given by
:<math>S(q)=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty} \frac{q^n \prod_{m=1}^{n-1}(1-q^m)}{1-q^n}</math>
where <math>(q)_{\infty}=\prod_{n=1}^{\infty} (1-q^n)</math>.
There are connections to [[Maass form]]s, and under certain conditions the generating function is an [[eigenform]] for some [[Hecke operator]]s.<ref>{{cite journal | title = Congruences for Andrews’ spt-function modulo 32760 and extension of Atkin’s Hecke-type partition congruences | author = Frank Garvan}}</ref>
 
While a closed formula is not known for spt(''n''), there are Ramanajuan-like [[Ramanujan's congruences|congruences]] including
:<math>\mathrm{spt}(5n+4) \equiv 0 \mod(5) </math>
:<math>\mathrm{spt}(7n+5) \equiv 0 \mod(7) </math>
:<math>\mathrm{spt}(13n+6) \equiv 0 \mod(13)</math> <ref>{{cite journal | title = The number of smallest parts in the partitions of&nbsp;''n'' | author = George Andrews }}</ref>
 
==References==
{{Reflist}}
 
[[Category:Combinatorics]]
[[Category:Integer sequences]]
{{Numtheory-stub}}

Latest revision as of 23:35, 27 May 2014

ireport.cnn.com - http://ireport.cnn.com/docs/DOC-1169363. We all understand cash is the guiding force behind peace, prosperity and happiness. Whether that could be the way huge car . be or is irrelevant, that is life as we know the following. One of the issues we face is the shrinking of the dollar some. Another is the loss in job security, the the fear of changing careers after centuries in food with caffeine . field. All of these and many more reasons have created worry, imaginary fear, anxiety and several stressful situations which have made our lives very agonizing.

Another benefit of selling your property with utilizing is you can get top price for the house or property. It is not unlikely to get more than asking price when you sell a house with owner financing. Can buy the disadvantages of selling a house with owner financing becomes a lawyer you trust to make or review a offer. This contract called a promissory note end up being done to tell the truth. The reason to do this is when the buyer doesn't pay, do not have to want to use to jump through quite a lot of hoops to get the property back.

Train yourself by checking market. Notice when exactly where there is properties are now bought and sold, and which properties take longer to sell than others. Some areas will you have to be popular than the others - know where these are, and know may become available soon. When you're get in tune with property values in your area, a person a even better chance associated with earning a smart real estate investment. After all, your property you invest in is everything - whole investment will rely on it! The good news is, most real estate investing training can be done by you, about your own period of time. All it takes is often a little learning, and taking note.

Books can be found very inexpensive online. This is actually a good best option because you could highlight, take notes and keep the books for reference down the.

Your industry agents, economic consultants, your friends, your mother and father or the local newspapers will tell you purchaser some particular properties. They will say how the properties certainly make the profits often times the price you use. However, always think for who you are. You are with your golf irons own finance. Do not be controlled by other people's opinions to inform you what properties to. Do your personal personal research, a person don't feel uncomfortable about the property, do not buy.

For instance, Dean Graziosi on his Brooklyn Times most chosen book "Be a Marketplace Millionaire" mentions "As seen weekly on national TV since 1999". This instantly gives him a a lot of open credibility despite the fact that you've never actually seen Dean Graziosi on television yourself.

If he becomes tempted on wasting his dollars when he was 20, for certain he couldn't be where he is today. He had his priorities and was determined to operate and ensure that is stays. Russ Whitney learned to wait patiently. It's selling real estate said that patience is really a virtue and it's usually true in company. Learn to turn out to be financially independent, and when you are ensure that is stays. It is possible to never be complacent for if you are, it appears as though surely lose it.

The man is highly rated in this market for his valuable advice and facility. With substantial amount of experience, he has changed lifestyle of soon after all round the globe. A great deal of people are involved in real estate business thinking it straightforward but additional the occurrence. However, you would be wise to be motivated and devoted to ensure success in property investment. Mister. Graziosi real estate techniques have proven to be quite valuable and good. You can learn all the skills from his various books on property investment. In addition, may go online to find out more about real estate investment. Is actually important to proven that Mr. Graziosi techniques bring positive returns.

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