A129894, patterns, randomness, and clumps

[Jon Schoenfield]

<<  The best I can say is to (1) get as much data as possible, and (2) use 
your best judgment.  >>

'Sounds like a good approach to me!  On the other hand, sometimes my best 
judgment isn't as good as I'd like....   :-/

In the case of A129894, I eventually realized that, yes, the numbers _are_ 
tending to "clump," and with good reason:  as n gets larger, the average 
value of the digits of n! -- other than the string of zeroes at the end --  
can reasonably be expected to tend toward (0+1+2+3+4+5+6+7+8+9)/10 = 9/2, 
and there are some ranges of values of n where it's much easier for the sum 
to come out to an exact multiple of n than it is in other ranges.

E.g., around n = 225000, the number of zeroes at the end should be almost 
n/4 = 56250 (I get 56248), and since 225000! has 1106528 digits, it'll have 
about 1050280 digits that precede the terminal string of zeroes, and 
1050280*9/2 = 4726260, which is very close to 21*n, so I shouldn't have been 
too surprised when I got up this morning and saw that the next terms after 
the clump in the 48000s were in a clump near 225000.  I should've realized 
this earlier....

But I do appreciate your comments on my broader question(s)!  :-)

Thanks again,

-- Jon

----- Original Message ----- 
From: <franktaw at netscape.net>
To: <jonscho at hiwaay.net>; <seqfan at ext.jussieu.fr>
Sent: Friday, June 08, 2007 11:06 AM
Subject: Re: A129894, patterns, randomness, and clumps


> No, these are not dumb questions at all.  I wish I had answers as good.
>
> The best I can say is to (1) get as much data as possible, and (2) use 
> your best judgment.
>
> One can't expect that every conjecture will turn out to be correct.  I try 
> for at least a 90% (subjective) confidence level.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Jon Schoenfield <jonscho at hiwaay.net>
>
> This got me thinking about a broader question:  How do you decide whether
> some observed behavior of a deterministic sequence -- some perceived
> pattern -- seems sufficiently significant that you think it's worth 
> offering
> a conjecture to your colleagues about that behavior?  Do you apply some 
> sort
> of statistical test(s)?  Is it more a matter of what some might call a 
> "gut
> feeling," based partly on what you know about the sequence thus far, and
> partly on prior experiences with a variety of sequences?  And does any
> confidence you might place in "gut feelings" about sequences in general 
> tend
> to get shaken when something very surprising turns up regarding some other
> problem, e.g., the discovery of "Numbers n such that ceiling(
> 2/(2^{1/n}-1) ) is not equal to floor( 2n/(log 2) )" (i.e., terms in
> A129935)?
>
> ... My apologies if these are obviously dumb newbie questions!  ?:-|
>
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