[seqfan] Re: semiprimes in step-6 arithmetic progression A125025

[hv at crypt.org]

"Richard J. Mathar" <mathar at mpia-hd.mpg.de> wrote:
:In the list of achievable lengths of arithmetic progressions of
:semiprimes, A125025, there still is a lower bound a(6), semiprime sequences 
:with gaps/step-size 6, that is conjectured.  [The bound is actually found by a 
:brute-force-attack checking semiprimes up to 11 million....]
:
:It may be useful to either confirm a(6), i.e., to proof that there is no
:sequence of 16 or more semiprimes with gap=6, or to truncate the sequence
:to a(1..5) and to add a(6)>=15, a(12)>=14, a(18)>=15, a(24)>=13.. as comments
:(the latter read from my PDF file in A124750).

To address your point here: it absolutely makes sense to me to truncate the
sequence and move best-known bounds on later values into the comments.

I'm also dubious about the other comment on the sequence:

:a(n) is at most A053669(n)^2, with equality if and only if A053669(n)^2
:is the first semiprime in the corresponding arithmetic progression.

I can easily see "with equality if and only if A053669(n)^2 appears as one
of the semiprimes in the corresponding arithmetic progression", but I don't
see why it would have to be the first. Of course there's almost certainly
some constant c (maybe 30?) such that A053669(n)^2 < n for all n > c, and
"first semiprime" could be justified if all n <= c had been manually
verified. But surely a comment would mention such complexities if they
were the case?

Hugo