partial_sums_of_lpf.sf

#!/usr/bin/ruby

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

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

# See also:
#   https://projecteuler.net/problem=521

func partial_sums_of_lpf(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)
}

for k in (1..7) {
    say "S(10^#{k}) = #{partial_sums_of_lpf(10**k)}"
}

__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
S(10^12) = 18435592284459044389811
S(10^13) = 1699246543196666002725979
S(10^14) = 157589263416765879793706013
S(10^15) = 14692398591103449239490624723