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

Douglas McNeil mcneil at hku.hk
Fri Sep 10 15:21:39 CEST 2010

Note that all of your last terms seem to end in 7 (correcting 117 to
177).. not the only way a sequence can end, but common enough.

Thinking out loud always ends in disaster for me, but:

Assume a term s=p+q ends with an odd digit d.  The next term would be
p*q, with p+q=d (mod 10).  But then one of p,q must be even, and one
must be odd.  Since p <= q are prime, p must be 2, and q=s-2, which is
pretty stringent.  Any term s ending in 7, for example, has q=5 mod
10, which for q > 5, can't be prime. E.g. in 6,9,14,33,62,177: 9
survives only because 7 is prime; 33 because 31 is prime; and 177
fails because 175 is composite.



Department of Earth Sciences
University of Hong Kong

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