[seqfan] Re: Chain of semiprimes: a(n)=least semiprime p*q such that a(n-1)=p+q.

[Vladimir Shevelev]

Zak,

In my opinion, these sequences are very interesting and, most likely, none of them to be infinite.
But 117 is not semiprime!

Best regards,
Vladimir

----- Original Message -----
From: zak seidov <zakseidov at yahoo.com>
Date: Friday, September 10, 2010 10:57
Subject: [seqfan] Chain of semiprimes: a(n)=least semiprime p*q such that a(n-1)=p+q.
To: seqfaneu <seqfan at seqfan.eu>

> Chain of semiprimes:  
> a(n) = the least semiprime p*q such that a(n-1)=p+q 
> (p<=q both prime). 
> 
> For a(1)=6 we have:
> 6,9,14,33,62,117.
> 
> (Notice that if we start with 4, we get the degenerate sequence 
> 4,4,4,4,4,4,4,4,... not in OEIS, but see
> A113311  Expansion of (1+x)^2/(1-x).) 
> 
> The last term is 117:
> there is no semiprime  p*q such that p+q=117!
> Hence the sequence is finite and consists of 6 terms:
> 6,9,14,33,62,117.
> 
> What about longer sequences?
> Here are some examples:
> 
> Sequence with a(1)=25 has 7 terms:
> 25,46,129,254,753,1502,4497.
> 
> Sequence with a(1)=133 has also 7 terms but has the very large 
> last term:
> 133,262,1285,2566,43333,86662,7186057:
> there is no semiprime  p*q such that p+q=7186057!
> 
> Next record sequence (9 terms with the largest last term) is
> 469,934,4645,9286,27849,55694,167073,334142,14366257.
> there is no semiprime  p*q such that p+q=14366257!
> 
> Yet another record: (10 terms with a very large - and unknown 
> yet - last term)
> 1654,27829,55654,1279513,2559022,181685521,
> 363371038,
> 10537759261,
> 21075518518,
> 2887346018197,?
> 
> Anyone may wish to continue the sequence 
> (Mmca is too slow)?
> Is this sequence finite?
> 
> Is, in general, sequence finite for any a(1)?
> 
> Thanks, Zak
> 
> 
> 
>       
> 
> 
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> 

 Shevelev Vladimir‎