A small factors sequence

[Creighton Dement]

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> Date: Wed,  7 Dec 2005 05:56:03 +0100
> Subject: Re: A small factors sequence
> From: "N. J. A. Sloane" <njas at research.att.com>
> To: seqfan at ext.jussieu.fr, "Gerald McGarvey"
> <Gerald.McGarvey at comcast.net>, "Creighton Dement" <crowdog at crowdog.de>

> I'm adding this to the OEIS!
> NJAS
> 
> %I A112403
> %S A112403
> 1,8,72,120,330,1584,1716,3432,12870,11440,19448,63648,50388, %T
> A112403 77520,232560,170544,245157,692208,480700,657800,1776060, %U
> A112403 1184040,1560780,4071600,2629575,3365856,8544096,5379616 %V
> A112403
> 1,-8,-72,-120,330,1584,1716,-3432,-12870,-11440,19448,63648,50388, %W
> A112403 -77520,-232560,-170544,245157,692208,480700,-657800,-1776060,
> %X A112403 -1184040,1560780,4071600,2629575,-3365856,-8544096,-5379616
> %N A112403 G.f.:
> (1-16*x+56*x^3-140*x^4+28*x^6-16*x^7+28*x^2+56*x^5+x^8)/(x^2-x+1)^8.
> %C A112403 Sequence is interesting because the initial terms seem to
> have many small factors.
> %O A112403 0,2
> %K A112403 sign,done
> %A A112403 Creighton Dement, Dec 06 2005
 
 

Dear Neil and Seqfans, 

Thanks for adding this sequence. It's just a guess, but it seems the
terms a(n) through, say, a(n+10)  have n as their approximate greatest
factor.  

Ex. Writing ( m[n], p ), let n stand for the largest factor of the m-th
term of the sequence. Let p be be the m-th term itself (disregarding
signs).  
Ex. a(15) = -(2)^4*(3)^2*(5)*(17)*(19) = -170544 so ( 15[19], 170544 )

Then 
( 50[53], 231917400 )
( 60[61], 1557578880 )
( 80[83], 5373200880 )
( 100[103], 24370067800 )
( 199[199], 2721949282900 )  Note: ( 200[103], 2817696242600 ) 
( 293[293], 39494993171634 ) 
( 401[401], 348499184786181 )
( 500[503], 1616297597409000 )
( 600[601], 11502809368023600 )
( 701[701], 17004576084168816 )
( 797[797], 41609248512914280 )
( 901[907], 97890610452628906 ) 
( 1100[1103], 394087856850630800 ) 

Note: a(900) =
(2)^4*(3)^2*(5)^2*(11)*(17)*(41)*(43)*(53)*(113)*(151)*(181)

Not presented here: Let m[[n]] stand for the second largest factor of
the m-th term of the sequence (and m[[[n]]] for the third largest
factor, etc.). Then it appears that m[n]/m[[n]] stays very close to 2 as
n increases. 

Lastly, the numerator of the g.f. looks a "bit prettier" when it is
written as 
(1-16*x+28*x^2+56*x^3-140*x^4+56*x^5+28*x^6-16*x^7+x^8)

Sincerely, 
Creighton


-It's a shame when the girl of your dreams would still rather be with
someone else when you're actually in a dream.