A113910 and Lesser of twin primes

[Creighton Dement]

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> Date: Mon, 30 Jan 2006 22:36:04 +0100
> Subject: A113910 and Lesser of twin primes
> From: "Creighton Dement" <crowdog at crowdog.de>
> To: seqfan at ext.jussieu.fr

> Dear Seqfans,
> 
> In reference to
> http://public.research.att.com/~njas/sequences/A113910
> 
> conjecture: a(n+4) = A001359(n+2) for all n.
> A001359:  Lesser of twin primes.
> 
> An immediate corollary of the above would be: 9 is the only composite
> term of the  sequence... perhaps this is somehow easier to prove.
> 
> Note: for n = 3, 4, 5, 6, 7, 8, 9, 10 ...
> 
> ((Lucas(n+1) - 2*A006206(n+2))/(A006206(n+2) - A006206(n))) = [3, 7,
> 5, 19/3, 31/4, 9, 87/10, 149/14, 11, 135/11, 663/50, 1094/77,
> 1787/120, 2939/181, 17, 7849/434, 12799/672, 20894/1041, 34031/1622,
> 55469/2514, 45131/1962, 146921/6115, 238915/9554, 194252/7465,
> 631347/23386, 1025917/36617, 29, 2706059/90178, 4393211/141710,
> 3565643/111405, 11573003/350702]
> 
> or
> 
> = [3.0, 7.0, 5.0, 6.333333333, 7.750000000, 9.0, 8.700000000,
> 10.64285714, 11.0, 12.27272727, 13.26000000, 14.20779221, 14.89166667,
> 16.23756906, 17.0, 18.08525346, 19.04613095, 20.07108549, 20.98088779,
> 22.06404137, 23.00254842, 24.02632870, 25.00680343, 26.02170127,
> 26.99679295, 28.01750553, 29.0, 30.00797312, 31.00141839, 32.00613078,
> 32.99953522}
> 
 
Another note on this:

Subtracting (i + 1) from the quotient 
((Lucas(n+1) - 2*A006206(n+2))/(A006206(n+2) - A006206(n))), 

gives

seq((Lucas[i+1] - 2*A[i+2])/(A[i+2]-A[i]) - i - 1  , i=1..200) 

1, 4, 1, 4/3, 7/4, 2, 7/10, 23/14, 1, 14/11, 63/50, 93/77, 107/120,
224/181, 1, 471/434, 703/672, 1115/1041, 1591/1622, 2675/2514,
1967/1962, 6276/6115, 9619/9554, 7627/7465, 23311/23386, 37258/36617, 1

Is there a simple and/or neat reason why both the numerators and
denominators in the the above reduced fractions appear to grow "rather
exponentially" - except for the points where they drop to 1? 

See Maple worksheet: (hit !!! to execute entire page)
http://www.crowdog.de/IntegerTwins.mw

Many thanks, 
Creighton