On age, perfection, semi-primality and being 142

Some years ago, the Frau was 28 years old: one day she became 29 and I cockily wrote in her birthday greeting "No longer perfect, but irreducible", and since that date have paid close attention to the primality or otherwise of our respective ages.

She was pleased and impressed when I noted that both before and after our recent birthdays (many thanks for your cards and gifts), both our ages were semiprime: which is to say all four numbers have precisely 2 prime factors (but you knew that already).

Which leads to the obvious questions:

1. What is the density of semiprimes?
2. What is the density of consecutive semiprimes: in particular is it asymptotically >0?

It took some web-searching to determine that an answer to the first question is x log log x / log x - a result I was unable to prove myself that is predictably due to Landau, although this is reported as a poor approximation, and better, less quotable, asymptotic formulae exist.

There does not seem to be a clear answer to the second question, which it need not surprise us pre-occupied the ubiquitous Paul Erdös (did I mention I once beat him at chess?). A simple computer program suggests there are many consecutives, and indeed triples (clearly, 4 in a row is no-go). Heath-Brown has shown there is an infinite number of such pairs [1] but I am unsure of their density.

Anyway, if I live to be 141-142, the pair of us can enjoy this happy numerological event again.

References

1. D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31, 141–149, 1984.