Another Weird Day

It's been 560 days since my last $\mathbf{\text{weird}}$ day. Today I turned $\mathbf{\text{27790}}$ days old and once again it's weird. I've written about this type of number before in blog posts titled Weird Numbers on August 4th 2019 and also in an earlier post titled Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers on November 22nd 2018.

Most abundant numbers are pseudoperfect meaning that a subset of their proper divisors can be chosen so that their sum is equal to the number. If the sum of all the proper divisors of a number equals the number itself, then the number is said to be perfect. The first few perfect numbers are 6, 28, 496, 8128 and 33550336. An example of an abundant number that is pseudoperfect is 24 with proper divisors of 1, 2, 3, 4, 6, 8 and 12. Adding these gives 36 so the number is clearly abundant. However, if we add the proper divisors without including 12, we reach 24 and so the number is pseudoperfect.

There are two types of weird numbers: primitive and non-primitive. A non-primitive weird number is a number that is a multiple of a weird number. The primitive weird numbers up to 40000 are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792 and 17272. Here is a list of all weird numbers (primitive and non-primitive) up to 40000:

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, 30170, 30310, 30730, 31010, 31430, 31990, 32270, 32410, 32690, 33530, 34090, 34370, 34930, 35210, 35630, 36470, 36610, 37870, 38290, 38990, 39410, 39830, 39970

All the numbers above with the exception of the primitive weird numbers in red are multiples of 70. Any weird number multiplied by a prime number that is greater than the sum of the divisors of the numbers is itself weird. Thus 70 has a sum of divisors of 144 and the first prime above this is 149. 70 x 149 produces the weird number 10430.

As can be seen, my next weird number (28070) is 280 days away and after that there is a gap of 560 days to my next weird number (28630). The gaps of course are determined by the distance between successive prime numbers. Thus:$$\begin{array}{rl}27790& =70\times 397\\ 28070& =70\times 401\\ 28630& =70\times 409\end{array}$$Here is a fuller list of primitive weird numbers (link):

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

While this post is a little repetitive, I feel it's important to acknowledge the occurrence of weird numbers when they occur because of their rarity.