$$2+3+5+17+97+599\cdots a_t \gt t!$$
What does that mean? Well it is a sum that follows specific rules. For one, the number of terms in the sequence is $t$. Similarly, $a_t$ represents the $t$'th term itself.
$$\underbrace{2+3+5}_{t=3}$$ $$2+3+\underbrace{5}_{a_3=5}$$
Also, note that the entire sum up to point $a_t$ MUST be $\gt t!$
$$2 \gt 1!$$ $$2+3 \gt 2!$$ $$2+3+5 \gt 3!$$ $$2+3+5+17 \gt 4!$$ $$\vdots$$ $$\text{and so on.}$$
A few final rules:
1) $a_t$ must be prime.
2) You must find the smallest prime that for each term that fits. An example of this would be the term $a_4$. $a_4$ can't be $19$ or $21$ because $17$ is smaller and fits all other descriptions. $13$, though smaller than $17$, does not make the sum $\gt$ $4!$ and thus is impossible to make as $a_4$.
3) You cannot add the same prime numbers twice, such as something you CANNOT do would be $2+2 \gt 2!$ because you would be repeating 2.
Now that you understand the rules of which the sum follows, I have a question about $a_t$. Although, there is no function $f$ of which $f(1)=a_1, f(2)=a_2, f(3)=a_3, f(4)=a_4...$ and so on, is there a function in which the only prime numbers generated by the function are $a_1,a_2\cdots,$ etc?
What do I mean by function though? Well this function $f$ produces two numbers, composite numbers and prime numbers. This function we are talking about is where whole numbers are inputted and whole numbers come out. If the only prime numbers you produce are $a_t$, then what is the function?
Thank you very much!
It seems that you need an additional rule to generate these terms: $a_m>a_n$ if $m>n.$
Of course there is a function $f$ which gives the terms, namely $f(n)=a_n.$ Perhaps you are asking about a function generated by some sort of closed-form formula, in which case you'd need to specify what restrictions this formula would be under.
This sequence is pretty easy to generate; for example, $a_{400}=\;$63874436539507335524192100393704248336075826568437952060037734827945391176263864357687782450844606514603614698598267168821177417476308641640889583675185990442498300015461067020395787496426278628714202576662549213436116980130822038626106449607775797166927250827771628823932770599297593113791509235583498808195284326389356296050534455705130436493715864261003644312853687392324527546079055861724494395190712825567745319820100296866623796943671957712476538980496593407214012056865542746151857573273551348222018306037731609352521018995733837341228699349046625222457638589804416836023330295553233052274555247448835867297818676846505918318342089037730862133100488213610344252088952806253854846315324552251380639696341549478581727619210311246061245231218111893164935809769909780480000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000521.
$a_{1000}=\;$401984872817016679770158731489080981733655489346503917911256110519508573886230391571452164278482435395679508621586998862573035206121813750342422505840274646406427079582986686158904804099461543787259726445044280844754100487034546557816859816551849794765337493550448600849782061864779957864202127035464156256946119779697080672138634504612481851705148155056874998205875217930035056129599272151470916171764709865868661558318740970536185172014389949294154989697365006608726162787419011122961451558877950327052425939824555946911952578862066195476963713453110731446401963315977809895281096271427792261421467876314486149727079851232301994200659129537706882983545402879668483651463568776409850010444489432879345265739508740511852930694742062220710437062662266627298049587078859371880693467789966598700215604588054790702484057045496974993365782621632201045537310949336544469387699777691820634164922185318272574815872646612184280090401864207745589396880231373784936080350711966641675919601829295772263615132504612417770586104179528022865348322848356768951993627109004070553744980427945145416989740870131991850277243347667698017190248969867730175174384246650313625164881674652614728725299829028435593255366204750119211142898624737230764767941314724905312820921917830312646128629207070514107535332868841058231608460243188694029590534268705610466320710762555956183996481167272777324292516822613219028927597764129186489176025395484057894192431526420041701134258832696982138287165663134087890530686293940408006027191587769388419063367218464255430308094113356780224067948444761839094574521024237401933615752707423016515010150692253381823287485465456431085240626543136193619639590731999544138435561984132547959958576737902653837703120842180771077840188392710150058616055740279147815502734733790478835837924185610777522553016253972224817724908832001899162462703790826167291103394723225461881721753587152684401637856109221483187036348294595863329594566924529974079280845009981594738187074456817944422710877907594792473690722919546079426508829178986591875987366056938929326262880959234591095014533513049866454675165329543876829543271944799118650279811768197855086482129541972245057699649135930876249096160487431408818662613727191146043375668400905222723758256899376151706330295047796923646972173492095149536415828049851942161471559057725718528000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010567.