weird numbers

A number

is called weird if it is abundant but not pseudoperfect, i.e., if the sum of its proper divisors is larger than $n$

, but there is not a subset of them whose sum is $n$

For example, 70, whose proper divisors are 1, 2, 5, 7, 10, 14, and 35 is abundant since

$\[1+2+5+7+10+14+35$ 70,\]">

but it not possible to obtain 70 adding a subset of them.

There are infinite weird numbers and S. Benkowski and P. Erdös have proved that their asymptotic density is positive.

According to Wenjie Fang, there are no odd weird numbers below $1.8\cdot10^{19}$ .

Every even number greater than 11634 can be written as the sum of weird numbers.

Sidney Kravitz has proved that, if $h\ge1$ , $p$ is a prime, and

$q = \frac{2^h\cdot p -p -1}{p+1-2^h}\,,$

is prime, then $2^{h-1}\cdot p\cdot q$ is weird.

The first weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870 more terms

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

A graph displaying how many weird numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Weird Number
Wikipedia, Weird number
OEIS, Sequence A006037

Weird numbers can also be... (you may click on names or numbers and on + to get more values)

aban 70 836 abundant 70 836 4030 5830 7192 7912 9272 10430 10570 10792 + 997430 997570 999670 admirable 70 836 4030 5830 7192 7912 10792 17272 45356 83312 + 254012 388076 786208 alternating 70 836 5830 9272 21490 25690 27230 29470 41090 45010 + 985670 985810 987490 amenable 836 7192 7912 9272 10792 17272 45356 73616 83312 91388 + 987316 992332 997348 apocalyptic 836 5830 7912 9272 10430 10570 10792 10990 11410 11690 + 28630 29330 29470 arithmetic 70 836 4030 5830 7192 7912 10430 10570 10792 10990 + 997430 997570 999670 binomial 70 congruent 70 4030 5830 7192 9272 10430 10792 10990 12110 12670 + 995470 997430 999670 Cunningham 357910 Curzon 13790 16310 16730 17570 18410 19670 21770 22190 25130 29330 + 973070 973490 977690 d-powerful 9272 295330 338572 345590 439390 537548 539744 627836 827470 828730 876964 decagonal 19390 dig.balanced 10570 12530 13370 13510 13790 15890 26110 30170 33530 35630 + 985810 995470 997570 eban 4030 esthetic 343210 evil 836 4030 7912 10430 10570 11410 11690 13790 13930 14770 + 994490 995470 997570 fibodiv 91070 Friedman 125090 194390 227570 255010 315490 357910 458710 466130 902510 972268 gapful 10430 10570 10990 11410 11690 12110 12530 12670 13370 13510 + 797930 798770 799610 happy 70 836 11410 12110 13370 13930 18410 19670 21770 24710 + 992332 993790 997430 Harshad 70 10792 11410 12110 13370 13510 16030 18410 22190 24710 + 978956 985670 994490 hoax 83312 338572 475684 565972 776644 791692 821788 912076 927124 iban 70 11410 12110 14770 17272 21770 23170 24710 42070 44170 + 742070 744170 770210 iccanobiF 836 idoneal 70 inconsummate 9272 10430 16310 16730 21770 34370 36470 36610 37870 39410 + 983570 983990 992390 interprime 10990 16310 21770 23170 24430 29330 30310 32690 41510 45010 + 983990 990430 997430 junction 11410 12110 16030 16310 16730 28630 29330 30310 31010 34930 + 982030 990430 991130 katadrome 70 76510 87430 98630 763210 854210 875210 954310 973210 975310 lonely 31430 magic 10990 magnanimous 70 modest 836 70630 90370 100030 130270 140210 200270 350630 560630 600110 700630 706930 801010 nialpdrome 70 43330 54110 64330 74410 76510 77210 77630 87430 96110 + 977410 992110 997430 oban 70 836 odious 70 5830 7192 9272 10792 10990 12110 12530 12670 13370 + 997348 997430 999670 pandigital 10570 13790 26110 30170 pentagonal 70 4030 29330 pernicious 70 5830 7192 9272 10792 12530 13370 13510 15890 16310 + 997348 997430 999670 prim.abundant 70 836 4030 5830 7192 7912 9272 10792 17272 45356 + 555616 682592 786208 self 7912 10570 23170 24430 30730 42070 43330 89810 91388 98630 + 992332 994490 997348 sliding 70 Smith 73616 388076 512468 527516 738188 768284 786208 828476 843524 933812 sphenic 70 super Niven 70 super-d 10570 11690 16310 19810 22190 24710 25690 26810 30310 32690 + 977690 982310 985810 uban 70 Ulam 7192 16030 19810 23590 26530 34090 37870 46270 52990 53830 + 971810 990710 992332 unprimeable 4030 10570 13370 15890 16730 17272 17990 19670 20510 24290 + 991130 992332 994490 untouchable 836 7192 7912 12670 13510 16030 16310 16870 17272 18830 + 995470 997010 997570 wasteful 70 836 4030 5830 7192 7912 9272 10430 10570 10792 + 997430 997570 999670 Zumkeller 70 836 4030 5830 7192 7912 9272 10430 10570 10792 + 98630 99610 99890 zygodrome 776644