list manipulation - Generate PrimePower counting function

Is there a way to generate a counting function for prime powers - i.e. to create a similar function to PrimePi, but including prime powers. The following will, of course, generate a list of these numbers:

Select[ Range[1000000], PrimePowerQ]

Answer

A naive approach would be this:

primePower[n_] := Count[ Range @ n, _?PrimePowerQ]

This function works well however it might be very inefficient for large n. It takes a bit to evaluate e.g.

primePower[10^6]

78734

which is only a little bigger than

PrimePi[10^6]

78498

The latter is much more efficient since it uses advanced algorithms for counting primes which exploit sparse caching and sieving techniques ( in documentation pages see Some Notes on Internal Implementation, then J. C. Lagarias, V. S. Miller and A. M. Odlyzko "Computing π(x): the Meissel-Lehmer method," Math. Comp., 44 (1985) 537-560, another interesting resources The Prime Pages: Prime Number Research, Records and Results).

Therefore we proceed along a different way making use of PrimePi to count prime powers, namely we need to sum Log[ Prime[k], n] (it measure how many times appears given prime with different powers up to n, e.g. for k == 3 there are 5, 25, 125, 625, ..., 5^Floor[ Log[ Prime[3], n]]) over every prime up to PrimePi[ Sqrt[n] // Floor] :

primePowerPi[n_] := PrimePi[n] + Sum[   Floor @ Log[ Prime[k], n] - 1, 
                                      { k, PrimePi[ Sqrt[n] // Floor]}]

This might be done as well with:

Floor @ Log[ Prime[ Range[ PrimePi[ Sqrt[n] // Floor]]], n]

e.g.

Total[ Floor @ Log[ Prime[ Range[ PrimePi[ Sqrt[10^5] // Floor]]], 10^5] - 1] == 
primePowerPi[10^5] - PrimePi[10^5]

primePowerPi[10^5] - PrimePi[10^5]

True

108

Now we can find immediately:

primePowerPi /@ { 10^6, 10^7}

{78734, 665134}

while

PrimePi[10^7]

664579

For some limitations of Prime and PrimePi domains this reference would be helpful What is so special about Prime?.