Problem Statement

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.

Let's start with the solution:

use v5.36;
use enum qw(False True);

sub table ($c, @V) { my $t = $c * (my $w = 5); ( sprintf( ('%'.$w.'d')x@V, @V) ) =~ s/.{1,$t}\K/\n/gr }

sub is_idoneal ($n) {
    LOOP:
    for my $a (1 .. $n) {
        for my $b ($a+1 .. $n) {
            last if $a*$b + $a + $b > $n;
            for my $c ($b+1 .. $n) {
                return False if $n == (my $sum = $a*$b + $b*$c + $c*$a);
                last if $sum > $n;
            }
        }
    }
    True
}

say table 10, grep { is_idoneal $_ } 1..1850;