694 Cube-full Divisors.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 10 February 2020
# https://github.com/trizen

# See also:
#   https://oeis.org/A036966
#   THE DISTRIBUTION OF CUBE-FULL NUMBERS, by P. SHIU (1990).

# https://projecteuler.net/problem=694

func powerful(n, k) {  # k-powerful numbers <= n

    var list = []

    func (m,r) {

        if (r < k) {
            list << m
            return nil
        }

        for a in (1 .. iroot(idiv(n,m), r)) {
            if (r > k) {
                a.is_coprime(m) || next
                a.is_squarefree || next
            }
            __FUNC__(m * a**r, r-1)
        }
    }(1, 2*k - 1)

    list.sort
}

func sum_of_cubefull_divisors_count(n) {

    var t = 0
    var s = n.isqrt

    var sqrt_cubefull = powerful(s, 3)

    for k in (sqrt_cubefull) {
        t += idiv(n,k)
    }

    var cubefull = powerful(n, 3)

    for (var k = 1 ; k <= s; ++k) {

        var w = idiv(n,k)
        var i = cubefull.end.bsearch_le {|j|
            cubefull[j] <=> w
        }

        var r = idiv(n, cubefull[i])
        r = s if (r > s)
        t += ((1+i) * (r - k + 1))
        k = r
    }

    t -= s*sqrt_cubefull.len

    return t
}

for n in (1..18) {
    say ("S(10^#{n}) = ", sum_of_cubefull_divisors_count(10**n))
}

__END__
S(10^1) = 11
S(10^2) = 126
S(10^3) = 1318
S(10^4) = 13344
S(10^5) = 133848
S(10^6) = 1339485
S(10^7) = 13397119
S(10^8) = 133976753
S(10^9) = 1339780424
S(10^10) = 13397833208
S(10^11) = 133978396859
S(10^12) = 1339784112539
S(10^13) = 13397841446161
S(10^14) = 133978415161307
S(10^15) = 1339784153146359
S(10^16) = 13397841534812404
S(10^17) = 133978415355411689