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

zak seidov zakseidov at yahoo.com
Fri Sep 10 09:27:57 CEST 2010

```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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