[seqfan] Re: conjectured largest number which is not the sum of an n-almost prime and a prime.

franktaw at netscape.net franktaw at netscape.net
Mon Mar 30 02:23:43 CEST 2009

If this is submitted, the title should be "Largest number which is not
the sum of an n-almost prime and a prime", not "Conjectured largest
....".  That all the values are conjectured belongs in a comment.

I would be reluctant to conjecture the value of a(6) below without
checking past 10^9 -- probably at least to 10^10.  Large gaps near the
end of these sequences seem to be the rule rather than the exception.

-----Original Message-----
From: Jonathan Post <jvospost3 at gmail.com>

Connecting these seqs:
A130588      Integers which are not the sum of a 3-almost prime and a
prime.
A146295      Integers which are not the sum of a 4-almost prime and a
prime.
A146296      Integers which are not the sum of a 5-almost prime and a
prime.
A146297      Integers which are not the sum of a 6-almost prime and a
prime.

we get the following... ALL of whose values are conjectures, although
checked through 10^9

a(n) is the conjectured largest number which is not the sum of an
n-almost prime and a prime.
n=2,3,4,5,6: a(n) = 10, 300, 60060, 3573570, 446185740

These are not as artificial as they seem, as they derive from Chen's
approach to proving Goldbach's conjecture.