Thread: The prime counting function View Single Post
2020-10-09, 22:20   #3
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

220708 Posts

Quote:
 Originally Posted by R2357 And, (probably the most time consuming part for this method) : we take away, for each of the primes above x (for x#), whose square is below x#, the number of primes that they can multiply by to give less than x#, in the case 7#, 11 can be multiplied up to 19, giving us 4 taken away (11*11, 13, 17, 19), and 13² takes away another number, Maybe an interesting idea...
It is an interesting idea indeed ...and a nice reinvention (not in a bad sense; people reinvent things all the time while they learn - which is very much like embryological parallelism. While one grows as a mathematical being, one repeats what the past evolution has already done).

I think this idea is ~188 years old. you see - for squareful numbers, whose moebius function is zero, you subtract them back because you've already added them in the first pass. But if you used moebius function, you would have "added them with weight 0" (i.e. skipped).