partial_sums_of_lcm_count_function.sf

#!/usr/bin/ruby

# Let f(n) be the number of couples (x,y) with x and y positive integers, x ≤ y and the least common multiple of x and y equal to n.

# Let a(n) = A007875(n), with a(1) = 1, for n > 1 (due to Vladeta Jovovic, Jan 25 2002):
#   a(n) = (1/2)*Sum_{d|n} abs(mu(d))
#        = 2^(omega(n)-1)
#        = usigma_0(n)/2

# This gives us f(n) as:
#   f(n) = Sum_{d|n} a(d)

# This script implements a sub-linear formula for computing partial sums of f(n):
#   S(n) = Sum_{k=1..n} f(k)
#        = Sum_{k=1..n} Sum_{d|k} a(d)
#        = Sum_{k=1..n} a(k) * floor(n/k)

# See also:
#   https://oeis.org/A007875
#   https://oeis.org/A064608
#   https://oeis.org/A182082

# Problem from:
#   https://projecteuler.net/problem=379

# Several values for S(10^n):
#   S(10^1)  = 29
#   S(10^2)  = 647
#   S(10^3)  = 11751
#   S(10^4)  = 186991
#   S(10^5)  = 2725630
#   S(10^6)  = 37429395
#   S(10^7)  = 492143953
#   S(10^8)  = 6261116500
#   S(10^9)  = 77619512018
#   S(10^10) = 942394656385
#   S(10^11) = 11247100884096

func usigma0_partial_sum (n) {      # O(sqrt(n)) complexity

    var total = 0

    var s = n.isqrt
    var u = idiv(n, s+1)

    var prev = squarefree_count(n)

    for k in (1..s) {
        var curr = squarefree_count(idiv(n, k+1))
        total += (prev - curr)*k
        prev = curr
    }

    u.each_squarefree {|k|
        total += idiv(n, k)
    }

    return total
}

func S(n) {
    dirichlet_sum(n,
        {|k| (k == 1) ? 1 : usigma0(k)/2  },        # a(n)
        { 1 },
        {|k| (usigma0_partial_sum(k)-1)/2 },        # partial sums of a(n)
        { _ }
    )
}

for k in (1..5) {
    say "S(10^#{k}) = #{S(10**k)}"
}