prog.sf

#!/usr/bin/ruby

func g(n) {
    is_prime(n) ? 1 : 0
}

func f(n) {
    n.divisors.sum{|d|
        g(d) * g(n/d)
    }
}

func a(n) {
    2*n.isqrt.primes.sum{|p|
        primepi(floor(n/p))
    } - primepi(n.isqrt)**2
}

func b(n) {
    2*semiprime_count(n) - primepi(n.isqrt)
}

func c(n) {
    (a(n) + primepi(n.isqrt))/2
}

say 100.of(f).accumulate
say 100.of(a)
say 100.of(b)

say a(10**9)
say b(10**9)

say semiprime_count(10**9)

#say ((a(10**9) + primepi(1e9.isqrt))/2)
say 30.of {|n| a(n) + primepi(n.isqrt) }

say ''

say 30.of(c)
say 30.of { .semiprime_count }

say c(10**9 + 7)
say semiprime_count(10**9 + 7)

__END__
Partial sums of A230595: sum of the Dirichlet convolution of the characteristic function of primes (A010051) with itself from 1 to n.

a(n) = Sum_{k=1..n} Sum_{d|k} A010051(d) * A010051(k/d).
a(n) = 2*Sum_{p prime <= sqrt(n)} A000720(floor(n/p)) - A000720(floor(sqrt(n)))^2.
a(n) = 2*A072000(n) - A000720(floor(sqrt(n))).


PARI

a(n) = my(s=sqrtint(n)); 2*sum(k=1, s, if(isprime(k), primepi(n\k), 0)) - primepi(s)^2;