It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*sum(A005867(i)/A002110(i), i = 0..T(sqrt(N))) This can show for example that at least .16*N numbers are prime less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010

These numbers are divisible by their digital roots, which makes the sequence a subsequence of A064807. - Ivan N. Ianakiev, Oct 08 2013

The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. p(n)# is divisible by 3 for n >= 2. - Christian Schulz, Oct 30 2013