Zumkeller Numbers

 I've been thinking about Zumkeller numbers. Here's a definition:

A number is a Zumkeller number if its divisors can be split into two separate sets that have the same sum. A key requirement for this to be possible is that the total sum of all divisors must be an even number. If the total sum were odd, you couldn't possibly split it into two equal integer sums.

Now the great majority of Zumkeller numbers are \( \textbf{even} \) numbers. In the range up to 100,000 there are 24362, comprising 24.362% of the range. However, there are only 208 \( \textbf{odd} \) Zumkeller numbers in that range, comprising 0.208%. It is suspected that every abundant number with an even sum of divisors is a Zumkeller number but this has not been proven. Any counterexample would have to be enormous—current searches have shown that if one exists, it must be greater than \(2×10^{13}\).

Now how many abundant numbers are \( \textbf{not} \) Zumkeller number because their sum of divisors is \( \textbf{odd} \)? In the range up to 100,000, there are 229 such numbers comprising OEIS A156903:

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 968, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4624, 4900, 5000, 5184, 5202, 5408, 5776, 5832, 6050, 6084, 6272, 6400, 6498, 7056, 7200, 7744, 7938, 8100, 8450, 8464, 8712, 9216, 9248, 9522, 9604, 9800, 10000, 10368, 10404, 10816, 11025, 11250, 11552, 11664, 12100, 12168, 12544, 12800, 12996, 13122, 13456, 14112, 14400, 15138, 15376, 15488, 15876, 16200, 16900, 16928, 17298, 17424, 18432, 18496, 19044, 19208, 19600, 19602, 20000, 20736, 20808, 21632, 22050, 22500, 23104, 23328, 23716, 24200, 24336, 24642, 25088, 25600, 25992, 26244, 26912, 27378, 28224, 28800, 28900, 30258, 30276, 30752, 30976, 31752, 32400, 33124, 33282, 33800, 33856, 34596, 34848, 36100, 36450, 36864, 36992, 38088, 38416, 39200, 39204, 39762, 40000, 41472, 41616, 43218, 43264, 43808, 44100, 45000, 46208, 46656, 46818, 47432, 48400, 48672, 49284, 50176, 50562, 51200, 51984, 52488, 52900, 53792, 53824, 54450, 54756, 56448, 56644, 57600, 57800, 58482, 59168, 60516, 60552, 61250, 61504, 61952, 62500, 62658, 63504, 64800, 66248, 66564, 66978, 67600, 67712, 69192, 69696, 70688, 70756, 71442, 72200, 72900, 73728, 73984, 76050, 76176, 76832, 78400, 78408, 79524, 80000, 80802, 81796, 82944, 83232, 84100, 85698, 86436, 86528, 87616, 88200, 89888, 90000, 90738, 92416, 93312, 93636, 94864, 95922, 96100, 96800, 97344, 98568, 99225

Of the above, 227 are even with only two numbers being odd. These are 11025 and 99225:

• \(11025 = 3^2 \times 5^2 \times 7^2 = 105^2\)
• \(99225 = 3^4 \times 5^2 \times 7^2\ = 945^2 \)

Even 70, the smallest "weird number" (an abundant number that is not the sum of any subset of its proper divisors), is a Zumkeller number. It's sum of divisors is 144 > 140 and thus it is abundant. It's divisors are {1, 2, 5, 7, 10, 14, 35, 70}. These can be split into {70, 2} and {1, 5, 7, 10, 14, 35} with both summing to 72. In fact, up to 100,000 at least, \( \text{all}\) of the weird numbers are Zumkeller numbers. Here are the weird numbers up to 100,000:

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, 40390, 41090, 41510, 41930, 42070, 42490, 42910, 43190, 43330, 44170, 44870, 45010, 45290, 45356, 45710, 46130, 46270, 47110, 47390, 47810, 48370, 49070, 49630, 50330, 50890, 51310, 51730, 52010, 52570, 52990, 53270, 53830, 54110, 55090, 55790, 56630, 56770, 57470, 57610, 57890, 58030, 58730, 59710, 59990, 60130, 60410, 61390, 61670, 61810, 62090, 63490, 63770, 64330, 65030, 65590, 65870, 66290, 66710, 67690, 67970, 68390, 68810, 69370, 69790, 70630, 70910, 71330, 71470, 72170, 72310, 72730, 73430, 73570, 73616, 74270, 74410, 74830, 76090, 76370, 76510, 76790, 77210, 77630, 78190, 78610, 79030, 80570, 80710, 81410, 81970, 82670, 83090, 83312, 83510, 84070, 84910, 85190, 85610, 86030, 86170, 86590, 87430, 88130, 89390, 89530, 89810, 90230, 90370, 90790, 91070, 91210, 91388, 91490, 92330, 92470, 92890, 95270, 95690, 96110, 96670, 97930, 98630, 99610, 99890

I've only written about Zumkeller numbers on one previous occasion. That was a post titled Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers that I uploaded in November of 2018. In that post I made reference to the person behind these numbers by quoting a eulogy by Neil Sloane, the originator of the OEIS.

I am deeply sorry to have to report that Reinhard Zumkeller passed away at the end of March 2016. He suffered from pancreatic cancer, which had already progressed to an advanced stage when it was diagnosed. He was a long-time contributor to the OEIS, and was later an editor and then a diligent and dedicated editor-in-chief. Between 2000 and 2016 he contributed over 23000 items to the OEIS. Reinhard was a great Haskell expert, and he was already ready to write a Haskell program and compute 10000 terms when I was studying a new sequence and wanted to see a graph. He will be greatly missed. Neil Sloane, July 3, 2016.