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admirable numbers
A number  $n$  is called admirable if there exists a divisor  $d$  of  $n$  such that  $2n = \sigma(n)-2d$. In other words,  $n$  is equal to the sum of its proper divisors, where one of them has minus sign.

For example, 12 is admirable because its proper divisors are 1, 2, 3, 4, 6, and 12 = 1 + 2 + 3 - 4 + 6.

Clearly, admirable numbers are a subset of abundant numbers and they are infinite because, for example, all the numbers  $6p$, with  $p>3$  prime, are admirable.

The largest number that cannot be written as a sum of admirable numbers is 1003.

Pairs of consecutive admirable numbers are rarer than pairs of consecutive abundant numbers. Up to  $10^{12}$  there are only two such pairs, namely (29691198404, 29691198405) and (478012798575, 478012798576).

Below, the spiral pattern of admirable numbers up to  $250^2$. See the page on prime numbers for an explanation and links to similar pictures.

spiral pattern of admirable numbers

The smallest 3 × 3 magic square made of admirable numbers is

282678102
174354534
60630426

The first admirable numbers are 12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222 more terms

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

A graph displaying how many admirable numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of admirable numbers
Useful links OEIS, Sequence A111592
Admirable numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 24 aban 12 20 24 + 99000966 abundant 12 20 24 + 49999974 Achilles 1372 476656 570375 alternating 12 30 54 + 98989854 amenable 12 20 24 + 99999956 apocalyptic 222 224 366 + 29994 arithmetic 20 30 42 + 9999942 betrothed 140 binomial 20 56 66 + 523776 c.nonagonal 491536 c.pentagonal 1266 1626 6126 + 97390806 c.triangular 1652176 8139856 12776464 + 86423536 cake 42 834 Catalan 42 compositorial 24 congruent 20 24 30 + 9999942 constructible 12 20 24 + 393222 Cunningham 24 120 224 + 86769224 Curzon 30 54 78 + 99999654 D-number 4095 d-powerful 24 224 1074 + 9996234 decagonal 3052 dig.balanced 12 42 56 + 95510716 double fact. 945 droll 672 eban 30 40 42 + 66064062 economical 224 1372 1504 + 8382464 equidigital 224 1372 1504 + 8382464 eRAP 20 24 esthetic 12 54 56 + 98765454 Eulerian 66 120 evil 12 20 24 + 99999844 factorial 24 120 fibodiv 366 5466 7806 + 44688364 Friedman 3378 12102 12964 + 983046 frugal 15872 122624 128768 + 67100672 gapful 120 140 1204 + 69989072 Gilda 78 364 3363582 Giuga 30 happy 70 368 464 + 9998574 harmonic 140 270 672 + 45532800 Harshad 12 20 24 + 99894896 heptagonal 7756 98704 2458672 13510575 hexagonal 66 120 491536 523776 highly composite 12 24 120 hoax 84 234 308 + 99994614 house 78 6308092 iban 12 20 24 + 777774 iccanobiF 836 idoneal 12 24 30 + 120 inconsummate 84 426 438 + 999858 interprime 12 30 42 + 99994614 Jordan-Polya 12 24 120 30240 junction 606 618 812 + 99999956 katadrome 20 30 40 + 76542 Leyland 54 368 945 lonely 120 lucky 4095 6435 9555 + 6220665 Lynch-Bell 12 24 672 + 4172 magic 87836 magnanimous 12 20 30 + 975772 metadrome 12 24 56 + 1234578 modest 222 618 654 + 99220336 Moran 42 84 114 + 99500212 Motzkin 6536382 narcissistic 9474 nialpdrome 20 30 40 + 99999844 nonagonal 24 474 nude 12 24 66 + 88828144 O'Halloran 12 20 84 140 oban 12 20 30 + 978 octagonal 40 11408 13736 odious 42 56 70 + 99999956 palindromic 66 88 222 + 8987898 pancake 56 1036 2212 + 98638036 panconsummate 12 20 24 + 78 pandigital 78 114 120 + 95510716 partition 30 42 56 + 614154 pentagonal 12 70 532 + 44045632 pernicious 12 20 24 + 9999856 Perrin 12 644 plaindrome 12 24 56 + 55577788 powerful 1372 476656 570375 practical 12 20 24 + 9989312 prim.abundant 12 20 30 + 99999956 primorial 30 pronic 12 20 30 + 67100672 Proth 7425 pseudoperfect 12 20 24 + 999942 repdigit 66 88 222 repunit 40 364 4095 Rhonda 15811444 Ruth-Aaron 24 78 104 + 110608 self 20 42 222 + 99999726 self-describing 14313318 14333116 15313314 + 33311514 sliding 20 70 650 Smith 438 654 762 + 99994614 sphenic 30 42 66 + 99999786 straight-line 222 234 246 + 678 strobogrammatic 88 906 80108 + 99696966 super Niven 12 20 24 + 1000002 super-d 318 1036 3342 + 9999668 superabundant 12 24 120 tau 12 24 40 + 45532800 tetrahedral 20 56 84 + 349504 tetranacci 56 triangular 66 78 120 + 523776 tribonacci 24 trimorphic 24 uban 12 20 30 + 99000042 Ulam 102 114 138 + 9999762 undulating 282 464 474 + 89898 unprimeable 532 534 894 + 9999892 untouchable 88 120 246 + 999786 upside-down 258 654 2828 + 89491612 vampire 229648 wasteful 12 20 24 + 9999942 weird 70 836 4030 + 786208 Zuckerman 12 24 224 672 Zumkeller 12 20 24 + 99988 zygodrome 66 88 222 + 99933222