521 Smallest prime factor.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 20 July 2020
# https://github.com/trizen

# Smallest prime factor
# https://projecteuler.net/problem=521

# Algorithm with sublinear time for computing:
#
#   Sum_{k=2..n} lpf(k)
#
# where:
#   lpf(k) = the least prime factor of k

# For each prime p < sqrt(n), we count how many integers k <= n have lpf(k) = p.

# We have G(n,p) = number of integers k <= n such that lpf(k) = p.
# G(n,p) can be evaluated recursively over primes q < p.

# Equivalently, G(n,p) is the number of p-rough numbers <= floor(n/p);

# There are t = floor(n/p) integers <= n that are divisible by p.
# From t we subtract the number integers that are divisible by smaller primes than p.

# The sum of the primes is p * G(n,p).
# When G(n,p) = 1, then G(n,p+r) = 1 for all r >= 1.

# Runtime: 3.742s (when Kim Walisch's `primesum` tool is installed).

local Num!USE_PRIMESUM = true
local Num!USE_PRIMECOUNT = false

func S(n) {

    var t = 0
    var s = n.isqrt

    s.each_prime {|p|
        t += p*p.rough_count(idiv(n,p))
    }

    t + sum_primes(s.next_prime, n)
}

say (S(1e12) % 1e9)

__END__
S(10^1)  = 28
S(10^2)  = 1257
S(10^3)  = 79189
S(10^4)  = 5786451
S(10^5)  = 455298741
S(10^6)  = 37568404989
S(10^7)  = 3203714961609
S(10^8)  = 279218813374515
S(10^9)  = 24739731010688477
S(10^10) = 2220827932427240957
S(10^11) = 201467219561892846337