prog.sf

#!/usr/bin/ruby

# Numbers of the form N = a+b+c such that N^3 = concat(a,b,c); a, b, c > 0.
# https://oeis.org/A328198

# Terms of A328198:
#   8, 45, 1611, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 19908, 55556, 60434, 77778, 422577, 427868, 461539, 478115, 488214, 494208, 543752, 559846, 598807, 664741, 757835, 791505, 807598, 4927940, 5555555, 6183170

# Cubes of the form N^3 = concat(a,b,c) with N = a+b+c;  a, b, c > 0.
# https://oeis.org/A328200

# Terms of A328200:
#   512, 91125, 4181062131, 87824421125, 93824221184, 121213882349, 128711132649, 162324571375, 171323771464, 368910352448, 7890107061312, 171471879319616, 220721185826504, 470511577514952, 75460133084214033, 78330233506116032, 98316229404133819, 109294197946170875

# Triples (a,b,c) such that (a+b+c)^3 = concat(a,b,c), a, b, c > 0, ordered by size of this value.
# https://oeis.org/A328199

func isok(n) {

    var N = n**3
    var d = N.digits.flip
    var e = d.end

    for i in (0 .. e) {

        var a = d.slice(0,i+1).join.to_i
        break if (a > n)

        for j in (i+1 .. e) {

            next if (d[i+1] == 0)
            var b = d.slice(i+1, j-i).join.to_i
            break if (b > n)

            for k in (j+1 .. e) {

                next if (d[j+1] == 0)
                var c = d.slice(j+1).join.to_i
                break if (c > n)

                var s = (a+b+c)

                if (s == n) {
                    say "cbrt(#{N}) = #{a} + #{b} + #{c} = #{s}"
                    return true
                }
                elsif (s > n) {
                    return false
                }
            }
        }
    }

    return false
}

for n in (1..1e6) {
    isok(n)
}

__END__
cbrt(512) = 5 + 1 + 2 = 8
cbrt(91125) = 9 + 11 + 25 = 45
cbrt(4181062131) = 418 + 1062 + 131 = 1611
cbrt(87824421125) = 878 + 2442 + 1125 = 4445
cbrt(93824221184) = 938 + 2422 + 1184 = 4544
cbrt(121213882349) = 1212 + 1388 + 2349 = 4949
cbrt(128711132649) = 1287 + 1113 + 2649 = 5049
cbrt(162324571375) = 1623 + 2457 + 1375 = 5455
cbrt(171323771464) = 1713 + 2377 + 1464 = 5554
cbrt(368910352448) = 3689 + 1035 + 2448 = 7172