Idoneal numbers

Idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x² ± Dy² (where x² is relatively prime to Dy²) is a prime power or twice a prime power.

A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integers a, b, and c with 0 < a < b < c.

There are only 65 known iodoneal numbers and is likely that no others exist. If there are others, it has been proven that there are at most, two more, and that no others exist below 1,000,000.

Task

Find and display at least the first 50 idoneal numbers (between 1 and 255).

Stretch

Find and display all 65 known idoneal numbers.

See also

Wikipedia: Idoneal numbers OEIS:A000926 - Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers)

Python

<syntaxheader lang="python"> Rosetta code task: rosettacode.org/wiki/Idoneal_numbers

def is_idoneal(num):

    Return true if num is an idoneal number 
   for a in range(1, num):
       for b in range(a + 1, num):
           if a * b + a + b > num:
               break
           for c in range(b + 1, num):
               sum3 = a * b + b * c + a * c
               if sum3 == num:
                   return False
               if sum3 > num:
                   break
   return True

row = 0 for n in range(1, 2000):

   if is_idoneal(n):
       row += 1
       print(f'{n:5}', end='\n' if row % 13 == 0 else )

</syntaxheader>

Output:

    1    2    3    4    5    6    7    8    9   10   12   13   15
   16   18   21   22   24   25   28   30   33   37   40   42   45
   48   57   58   60   70   72   78   85   88   93  102  105  112
  120  130  133  165  168  177  190  210  232  240  253  273  280
  312  330  345  357  385  408  462  520  760  840 1320 1365 1848

Raku

First 60 in less than 1/2 second. The remaining 5 take another ~5 seconds.

sub is-idoneal ($n) {
    my $idoneal = True;
    I: for 1 .. $n -> $a {
        for $a ^.. $n -> $b {
            last if $a × $b + $a + $b > $n; # short circuit
            for $b ^.. $n -> $c {
                $idoneal = False and last I if (my $sum = $a × $b + $b × $c + $c × $a) == $n;
                last if $sum > $n; # short circuit
            }
        }
    }
    $idoneal
}

$_».fmt("%4d").put for (1..1850).hyper(:32batch).grep( &is-idoneal ).batch(10)

Output:

   1    2    3    4    5    6    7    8    9   10
  12   13   15   16   18   21   22   24   25   28
  30   33   37   40   42   45   48   57   58   60
  70   72   78   85   88   93  102  105  112  120
 130  133  165  168  177  190  210  232  240  253
 273  280  312  330  345  357  385  408  462  520
 760  840 1320 1365 1848