What's the Next Term II

By -
Well, I don’t think I’m going to suck in anyone who hasn’t been heard from by now…

So, OK, what is the next term of this sequence? 0, 1, 2, 8.252 x 10^2466, ...

Here, "10^2466" means "ten to the 2466th power." The corresponding term is necessarily slightly approximate, since it contains 2467 decimal digits.

Here’s my answer (the one that the problem comes with): t(n) = n(!^n).

0 = 0
1! = 1
2!! = 2! = 2
3!!! = 6!! = 720! = 8.252 x 10^2466
4!!!! is a rather large number in most contexts

Hey, you wanna see a really big number? Howsabout (10^10)(!^(10^10))? True, it’s still only 0% of aleph-null.

Comments

  1. gcallah7:51 AM

    "4!!!! is a rather large number in most contexts."

    But not that of, "Shots of well vodka consumed by Wabulon at B61.

    ReplyDelete
  2. LarryRuane12:21 PM

    Are you sure about 3!!!? Using linux bc -l funcs.bc, both "factorial(720)" and "factorial(factorial(factorial(3)))" produce:

    26012189435657951002049032270810436111915218750169457857275418378508\
    35631156947382240678577958130457082619920575892247259536641565162052\
    01587379198458774083252910524469038881188412376434119195104550534665\
    86162432719401971139098455367272785370993456298555867193697740700037\
    00430783758997420676784016967207846280629229032107161669867260548988\
    44551425719398549944893959449606404513236214026598619307324936977047\
    76060676806701764916694030348199618814556251955925669188308255149429\
    47596537274845624628824234526597789737740896466553992435928786212515\
    96748322097602950569669992728467056374713753301924831358707612541268\
    34158601294475660114554207495899525635430682886346310849656506827715\
    52996256790845235702552186222358130016700834523443236821935793184701\
    95651072978180435417389056072742804858399591972902172661229129842051\
    60675790362323376994539641914751755675576953922338030568253085999774\
    41675784352815913461340394604901269542028838347101363733824484506660\
    09334848444071193129253769465735433737572477223018153403264717753198\
    45373414786743270484579837866187032574059389242157096959946305575210\
    63203263493209220738320923356309923267504401701760572026010829288042\
    33560664308988871029738079757801305604957634283868305719066220529117\
    48225105366977566030295740433879834715185526028053338663571391010463\
    36419769097397432285994219837046979109956303389604675889865795711176\
    56667003915674815311594398004362539939973120306649060132531130471902\
    88984918562037666691644687911252491937544258458950003115616829743046\
    41142538074897281723375955380661719801404677935614793635266265683339\
    50976000000000000000000000000000000000000000000000000000000000000000\
    00000000000000000000000000000000000000000000000000000000000000000000\
    00000000000000000000000000000000000000000000000

    which has around 1746 digits, not 2466. Great puzzle though; thanks for posting it.

    ReplyDelete
  3. gcallah12:38 PM

    I get the same answer on Max OS X bc -- which, of course, is the same program, so no surprise. The question I have is, "Do we have reason to think bc is reliable for such huge numbers?"

    (Of course, another question is, "Do we have reason to think Wabulon is reliable for such huge numbers?")

    ReplyDelete
  4. gcallah12:42 PM

    Larry, is there a way to force bc to output in scientific notation?

    ReplyDelete
  5. LarryRuane1:26 PM

    Yes, funcs.bc provides it (and at least on Linux that's what implements factorial(), so I assume you have it). For example:

    engnot(factorial(720), 5)
    2.60121*10^1746

    ReplyDelete
  6. Wabulon2:12 PM

    Err, uhh, I used Stirling's Formula (quite accurate for approximating large factorials), log10-ing everything. And, no, I am not particularly good at arithmetic (although I have a nice essay on Peano's Postulates if anyone wants to see it). And, of course, I really don't much care what the actual value of 720! is, it's the principle of the thing...

    ReplyDelete
  7. gcallah3:59 PM

    Yes, but Wabulon, how are we supposed to guess the formula if one of the items in the series is wrong?

    ReplyDelete
  8. Wabulon4:21 PM

    Larry -

    I used Stirling's formula in the form: log n! approximately =
    (n+0.5) log n - 0.4343n + 0.3991.
    Those coefficients checked out when I recalculated them, but, obviously, my arithmetic didn't, cuz when I redid it for 720!, I got roughly 2x10^1746. Sorry.

    ReplyDelete

Post a Comment

Popular posts from this blog

Libertarians, My Libertarians!

By -
Cruel to be kind means that I love you . Because, while I think you are mistaken, your hearts are in the right place -- yes, even you, Silas --  unlike some people . This Breitbart fellow (discussed in the link above), by all appearances, deliberately doctored a video of Shirley Sherrod to make her remarks appear virulently racist, when they had, in fact, the opposite import. I heard that at a recent Austrian conference, some folks were talking about "Callahan's conservative turn." While that description is not entirely inaccurate, I must say that a lot of these people who today call themselves conservative give me the heebie-jeebies.
Read more

"Pre-Galilean" Foolishness

By -
 I am currently reading The Master and His Emissary , which appears to be an excellent book. ("Appears" because I don't know the neuroscience literature well enough to say for sure, yet.) But then on page 186 I find: "Asking cognition, however, to give a perspective on the relationship between cognition and affect is like asking astronomer in the pre-Galilean geocentric world, whether, in his opinion, the sun moves round the earth of the earth around the sun. To ask a question alone would be enough to label one as mad." OK, this is garbage. First of all, it should be pre-Copernican, not pre-Galilean. But much worse is that people have seriously been considering heliocentrism for many centuries before Copernicus. Aristarchus had proposed a heliocentric model in the 4th-century BC. It had generally been considered wrong, but not "mad." (And wrong for scientific reasons: Why, for instance, did we not observe stellar parallax?) And when Copernicus propose...
Read more