O'Halloran numbers

O'Halloran numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

For this task, the term "cuboid" means a 3-dimensional object with six rectangular faces, where all angles are right angles, where opposite faces are equal, and where each dimension is a positive integer unit length.

The surface area of a cuboid so-defined is two times the length times the width, plus two times the length times the height, plus two times the width times the height. For example, a cuboid with l = 2, w = 1 h = 1 has a surface area of 10:

   2 × ( 2 × 1 + 1 × 1 + 1 × 2 ) = 10

The minimum surface area a cuboid may have is 6 - namely one for which the l, w, and h measurements are all 1:

   2 × ( l × w + w × h + h × l )
   2 × ( 1 × 1 + 1 × 1 + 1 × 1 ) = 6

Notice that the total surface area of a cuboid is always an integer and is always even, but there are many even integers which do not correspond to the area of a cuboid. For example, there is no cuboid with a surface area of 8.

Task

• Find and display the sixteen even integer values larger than 6 (the minimum cuboid area) and less than 1000

that can not be the surface area of a cuboid.

See also

• Numbers Aplenty - O'Halloran numbers
• OEIS:A072843 - O'Halloran numbers: even integers which cannot be the surface area of a cuboid with integer-length sides
• Idoneal numbers

ALGOL 68

BEGIN # find O'Halloran numbers - numbers that cannot be the surface area of  #
      #                           a cuboid with integer dimensions            #
    INT count        := 0;
    INT max area      = 1 000;
    INT half max area = max area OVER 2;
    FOR n FROM 8 BY 2 TO max area DO
        BOOL ohalloran := TRUE;
        FOR l TO half max area WHILE ohalloran DO
            FOR b TO half max area WHILE INT lb = l * b;
                                         lb < n AND ohalloran
            DO
                FOR h TO half max area WHILE INT bh = b * h, lh = l * h;
                                             INT sum = 2 * ( lb + bh + lh );
                                             sum <= n
                                             AND ( ohalloran := sum /= n )
                DO SKIP OD
            OD
        OD;
        IF ohalloran THEN
            print( ( " ", whole( n, 0 ) ) );
            count +:= 1
        FI
    OD
END

Output:

 8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924

J

   require'stats'
   2*(3}.i.501)-.+/1 */\.(|:3 comb 42)-i:1
8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924

Here, we use combinations with repetitions to generate the various relevant cuboid side lengths. Then we multiply all three pairs of these side length combinations and sum the pairs. Then we remove these sums from the sequence 3..500, and finally we multiply the remaining 16 values by 2.

Julia

""" Rosetta code task: rosettacode.org/wiki/O%27Halloran_numbers """

const max_area, half_max = 1000, 500
const areas = trues(max_area)

areas[1:2:max_area] .= false

for i in 1:max_area
    for j in 1:half_max
        i * j > half_max && break
        for k in 1:half_max
            area = 2 * (i * j + i * k + j * k)
            area > max_area && break
            areas[area] = false
        end
    end
end

println("Even surface areas < $max_area NOT achievable by any regular integer-valued cuboid:\n",
    [n for n in eachindex(areas) if areas[n]])

Output:

Even surface areas < 1000 NOT achievable by any regular integer-valued cuboid:
[2, 4, 8, 12, 20, 36, 44, 60, 84, 116, 140, 156, 204, 260, 380, 420, 660, 924]

Perl

use v5.36;
my @A;
my $threshold = 10_000; # redonkulous overkill

for my $x (1..$threshold) {
    X: for my $y (1..$x) {
        last X if $x*$y > $threshold;
        Y: for my $z (1..$y) {
           last Y if (my $area = 2 * ($x*$y + $y*$z + $z*$x)) > $threshold;
           $A[$area] = "$x,$y,$z";
        }
    }

say 'Even integer surface areas NOT achievable by any regular, integer dimensioned cuboid';
for (0..$#A) {
    print "$_ " if $_ < $threshold and $_ > 6 and 0 == $_ % 2 and not $A[$_];
}

Output:

Even integer surface areas NOT achievable by any regular, integer dimensioned cuboid:
8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924

Phix

Since we are going to check {1,1,2}, there is no point checking {1,2,1} or {2,1,1} etc.

with javascript_semantics
constant max_area = 1000, half_max = max_area/2
sequence areas = repeat(0,5)&tagset(max_area,6)

for x=1 to max_area do
    if odd(x) then areas[x] = 0 end if
    for y=x to floor(half_max/x) do
        for z=y to half_max do
            atom area = 2 * (x * y + x * z + y * z)
            if area > max_area then exit end if
            areas[area] = 0
        end for
    end for
end for

printf(1,"Even surface areas < %d NOT achievable by any regular integer-valued cuboid:\n%s\n",
         {max_area,join(filter(areas,"!=",0),fmt:="%d")})

Output:

You can also set max_area to 1,000,000 and get no more results.

Even surface areas < 1000 NOT achievable by any regular integer-valued cuboid:
8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924

PROMAL

;;; find O'Halloran numbers - numbers that cannot be the surface area of
;;;                           a cuboid with integer dimensions
PROGRAM OHalloran
INCLUDE LIBRARY

CON WORD maxArea = 1000
WORD count
WORD halfMaxArea
WORD n
WORD l
WORD b
WORD h
WORD lb
WORD lh
WORD bh
WORD sum
BYTE isOHalloran
BEGIN
count = 0
halfMaxArea = maxArea / 2
n = 8
WHILE n <= maxArea
  isOHalloran = 1
  l = 1
  REPEAT
    b  = 1
    lb = l
    REPEAT
      h = 1
      REPEAT
        bh  = b * h
        lh  = l * h
        sum = 2 * ( lb + ( h * ( b + l ) ) )
        IF sum = n
          isOHalloran = 0
        h = h + 1
      UNTIL h > halfMaxArea OR sum > n OR NOT isOHalloran
      b  = b + 1
      lb = l * b
    UNTIL b > halfMaxArea OR lb >= n OR NOT isOHalloran
    l = l + 1
  UNTIL l > halfMaxArea OR NOT isOHalloran
  IF isOHalloran
    OUTPUT " #W", n
    count = count + 1
  n = n + 2
END

Output:

 8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924

Raku

my @Area;

my $threshold = 1000; # a little overboard to make sure we get them all

for 1..$threshold -> $x {
    for 1..$x -> $y {
        last if $x × $y > $threshold;
        for 1..$y -> $z {
           push @Area[my $area = ($x × $y + $y × $z + $z × $x) × 2], "$x,$y,$z";
           last if $area > $threshold;
        }
    }
}

say "Even integer surface areas NOT achievable by any regular, integer dimensioned cuboid:\n" ~
   @Area[^$threshold].kv.grep( { $^key > 6 and $key %% 2 and !$^value } )»[0];

Output:

Even integer surface areas NOT achievable by any regular, integer dimensioned cuboid:
8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924

Wren

import "./seq" for Lst

var found = []
for (l in 1..497) {
    for (w in 1..l) {
        var lw = l * w
        if (lw >= 498) break
        for (h in 1..w) {
            var sa = (lw + w*h + h*l) * 2
            if (sa < 1000) found.add(sa) else break
        }
    }
}
var allEven = (6..998).where { |i| i%2 == 0 }.toList
System.print("All known O'Halloran numbers:")
System.print(Lst.except(allEven,found))

Output:

All known O'Halloran numbers:
[8, 12, 20, 36, 44, 60, 84, 116, 140, 156, 204, 260, 380, 420, 660, 924]

XPL0

int  L, W, H, HA, I;
char T(1000/2);                 \table of half areas
[for I:= 0 to 1000/2-1 do
        T(I):= true;            \assume all are O'Halloran numbers
for L:= 1 to 250 do
    for W:= 1 to 250/L do
        for H:= 1 to 250/L do
            [HA:= L*W + L*H + W*H;
            if HA < 500 then    \not an O'Halloran number
                T(HA):= false;
            ];
for I:= 6/2 to 1000/2-1 do
    if T(I) then
        [IntOut(0, I*2);  ChOut(0, ^ )];
]

Output:

8 12 20 36 44 60 84 116 140 156 204 260 380 420 660 924