[seqfan] A339709

[israel at math.ubc.ca]

I'd like to draw your attention to A339709, a joint submission by 
J. M. Bergot and me.  The title is

a(n) is the least even number that has exactly n decompositions as the sum 
of an odd prime and a semiprime, or 0 if there is no such number.

We conjecture that there is no even number with exactly 12 such 
decompositions (so a(12) should be 0), while a(n) > 0 for every other 
nonnegative integer.

Since a(12)=0 is only a conjecture, the Data stops with a(11) = 92, and I 
put the conjectured a(12) with values for all other n up to 800 in an 
A-file.

Is there any prospect at all of a proof that a(12)=0? The only plausible 
way I can think of would be some explicit estimates related to Chen's 
theorem that every sufficiently large even number is the sum of a prime and 
the product of at most two primes, to show that each even number greater 
than some N has more than 12 decompositions.

Cheers,
Robert