Weird Numbers

Today I turned 25690 days old and this number is weird. Every weird number is abundant, meaning that the sum of its proper divisors exceeds the number itself. In the case of 25690, its proper divisors are 1, 2, 5, 7, 10, 14, 35, 70, 367, 734, 1835, 2569, 3670, 5138 and 12845. These sum to 27302. For an abundant number, there is generally a subset of the proper divisors whose sum equals the number. In such cases, the number is described as pseudoperfect or semiperfect. For a perfect number, such as 6, the sum of the proper divisors (1, 2 and 3) equals the number. In rare instances, there is no subset of proper divisors that sum to the number and in such cases the number is described as weird.

Here is the list of weird numbers below 30000:

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, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430, 24710, 25130, 25690, 26110, 26530, 26810, 27230, 27790, 28070, 28630, 29330, 29470

Coincidentally, I'm 70 years of age at the moment and 70 is the first weird number. So today I'm a weird number of days old and a weird number of years old!

As can be seen, weird numbers are not all that frequent and in fact there are only 57 such numbers up to 30000, representing a frequency of about 0.19%. The numbers shown are all even. It is not known if there are any odd weird numbers but if there are, it's been shown that they must be very large.

I've mentioned weird numbers before, in my post Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers where I wrote on Thursday, 22nd November 2018:

While nearly all abundant numbers are pseudoperfect, some aren't. These numbers are termed weird and comprise OEIS A006037: weird numbers - abundant (A005101) but not pseudoperfect (A005835). From the comments to this sequence in the OEIS, we find:

Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers (A005101) and weird numbers (A006037), since the remaining integers are perfect numbers (A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers (A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790.

There are other interesting facts mentioned in the comments, including:

• The first weird number that has more than one decomposition of its divisors set into two subsets with equal sum (and thus is not a member of A083209) is 10430:

  1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215

  2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215.

• A weird number \(n\) multiplied with a prime \(p> \sigma(n) \) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor.
• No odd weird number exists below \(10^{21}\).

The primitive weird numbers up to 30000 are:

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272

From this we can note that 25690 = 70 x 367 where 367 \(> \sigma(70)\) = 144. The next weird number 26110 = 70 x 373.