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idoneal numbers
Euler's idoneal numbers, are those positive integers  $d$  such that if  $n$  can be written in only one way as  $x^2 \pm dy^2$  (where  $x^2$  is relatively prime to  $dy^2$) then  $n=p^k$  or  $p=2p^k$, where  $k\ge1$  and  $p$  is a prime.

An equivalent simpler definition is: a number is idoneal if and only if it cannot be written as  $ab + bc + ac$ for  $ab>c>0$">.

Sometimes they are called suitable numbers or convenient numbers.

Only 65 idoneal number are known and the list is conjectured to be complete. P.Weinberger proved in 1973 that the list can contain at most one more number.

The known idoneal numbers are 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.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of idoneal numbers

A graph displaying how many idoneal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of idoneal numbers
Useful links Mathworld, Idoneal Number
Wikipedia, Idoneal number
OEIS, Sequence A000926
Idoneal numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 ABA 18 24 72 aban 10 12 13 + 840 abundant 12 18 24 + 1848 Achilles 72 admirable 12 24 30 + 120 alternating 10 12 16 + 385 amenable 12 13 16 + 1848 apocalyptic 312 840 arithmetic 13 15 21 + 1848 astonishing 15 automorphic 25 Bell 15 betrothed 48 binomial 10 15 21 + 1365 brilliant 10 15 21 + 253 c.heptagonal 22 253 c.nonagonal 10 28 190 253 c.octagonal 25 c.pentagonal 16 c.square 13 25 85 c.triangular 10 85 760 cake 15 42 93 + 232 Catalan 42 Chen 13 37 compositorial 24 congruent 13 15 21 + 1848 constructible 10 12 15 + 408 Cullen 25 385 Cunningham 10 15 24 + 1848 Curzon 18 21 30 + 1365 cyclic 13 15 33 + 345 D-number 15 21 33 + 177 d-powerful 24 357 decagonal 10 85 232 deficient 10 13 15 + 1365 dig.balanced 10 12 15 + 520 double fact. 15 48 105 droll 72 240 Duffinian 16 21 25 + 385 eban 30 40 42 60 economical 10 13 15 + 177 emirp 13 37 emirpimes 15 58 85 + 177 equidigital 10 13 15 + 177 eRAP 24 esthetic 10 12 21 + 345 Eulerian 57 120 evil 10 12 15 + 1848 factorial 24 120 fibodiv 28 Fibonacci 13 21 Friedman 25 gapful 105 120 130 + 1365 Gilda 78 330 Giuga 30 good prime 37 happy 10 13 28 + 280 harmonic 28 Harshad 10 12 18 + 1848 heptagonal 18 112 hex 37 hexagonal 15 28 45 + 190 highly composite 12 24 48 + 840 hoax 22 58 85 Hogben 13 21 57 + 273 house 78 hyperperfect 21 28 iban 10 12 21 + 1320 iccanobiF 13 impolite 16 inconsummate 840 interprime 12 15 18 + 1320 Jacobsthal 21 85 1365 Jordan-Polya 12 16 24 + 240 junction 105 210 408 1320 Kaprekar 45 katadrome 10 21 30 + 840 Lehmer 15 85 133 Leyland 57 177 lonely 120 Lucas 18 lucky 13 15 21 + 1365 Lynch-Bell 12 15 24 + 312 magic 15 magnanimous 12 16 21 + 130 metadrome 12 13 15 + 357 modest 13 133 Moran 18 21 42 + 190 Motzkin 21 nialpdrome 10 21 22 + 840 nonagonal 24 nude 12 15 22 + 1848 O'Halloran 12 60 oban 10 12 13 + 760 octagonal 21 40 133 + 408 odious 13 16 21 + 385 palindromic 22 33 88 232 pancake 16 22 37 232 panconsummate 10 12 15 + 85 pandigital 15 21 78 + 210 partition 15 22 30 + 385 pentagonal 12 22 70 + 330 perfect 28 pernicious 10 12 13 + 520 Perrin 10 12 22 Pierpont 13 37 plaindrome 12 13 15 + 357 power 16 25 powerful 16 25 72 practical 12 16 18 + 1848 prim.abundant 12 18 30 + 102 prime 13 37 primeval 13 37 primorial 30 210 pronic 12 30 42 + 462 Proth 13 25 33 + 385 pseudoperfect 12 18 24 + 1848 repdigit 22 33 88 repfigit 28 repunit 13 15 21 + 1365 Ruth-Aaron 15 16 24 + 105 Saint-Exupery 60 self 42 165 312 + 760 self-describing 22 semiprime 10 15 21 + 253 sliding 25 70 133 520 Smith 22 58 85 sphenic 30 42 70 + 385 square 16 25 star 13 37 253 straight-line 210 345 357 840 strobogrammatic 88 strong prime 37 super Niven 10 12 24 + 1320 super-d 105 190 462 1848 superabundant 12 24 48 + 840 tau 12 18 24 + 240 tetrahedral 10 120 165 tetranacci 15 triangular 10 15 21 + 253 tribonacci 13 24 trimorphic 24 25 truncatable prime 13 37 twin 13 uban 10 12 13 + 93 Ulam 13 16 18 + 273 undulating 232 unprimeable 840 untouchable 88 120 210 + 520 upside-down 28 37 357 wasteful 12 18 22 + 1848 weak prime 13 weird 70 Zeisel 105 Zuckerman 12 15 24 + 312 Zumkeller 12 24 28 + 1848 zygodrome 22 33 88