Abundant But Not Zumkeller

A property of the number \( \textbf{28062}\) associated with my diurnal age today prompted me to look more closely at abundant numbers that are not Zumkeller. 28062 has the following properties:$$ \begin{align} 28062 &=2 \times 3^2 \times 1559\\ \text{ divisors }  &\rightarrow 1, 2, 3, 6, 9, 18, 1559, 3118, 4677, 9354, 14031, 28062 \end{align}$$Normally, for abundant numbers, the set of divisors can be divided into two mutually exclusive sets whose elements sum to the same number. However, this simply can't be done with 28062.

Now up to 40000, there are 718 abundant numbers that are abundant but not Zumkeller. 146 of these have an odd sum of divisors and an even split is not possible. The other 572 have an even sum of divisors and thus have the potential for an even split but it proves impossible to find one. Here are the details:

There are 572 abundant numbers with an \( \textbf{even}\) sum of divisors. They are (up to 40000):

[738, 748, 774, 846, 954, 1062, 1098, 1206, 1278, 1314, 1422, 1494, 1602, 1746, 1818, 1854, 1926, 1962, 2034, 2286, 2358, 2466, 2502, 2682, 2718, 2826, 2934, 3006, 3114, 3222, 3258, 3438, 3474, 3492, 3546, 3582, 3636, 3708, 3798, 3852, 3924, 4014, 4068, 4086, 4122, 4194, 4302, 4338, 4518, 4572, 4626, 4716, 4734, 4842, 4878, 4932, 4986, 5004, 5058, 5094, 5274, 5364, 5436, 5526, 5598, 5634, 5652, 5706, 5868, 5958, 6012, 6066, 6228, 6246, 6282, 6354, 6444, 6462, 6516, 6606, 6714, 6822, 6876, 6894, 6948, 7002, 7092, 7146, 7164, 7218, 7362, 7542, 7544, 7578, 7596, 7758, 7794, 7902, 7974, 8028, 8082, 8172, 8226, 8244, 8298, 8334, 8388, 8406, 8604, 8622, 8676, 8766, 8838, 8982, 9036, 9054, 9162, 9252, 9378, 9414, 9468, 9684, 9738, 9756, 9846, 9972, 10026, 10116, 10134, 10184, 10188, 10242, 10278, 10386, 10548, 10566, 10674, 10782, 10818, 10926, 11034, 11052, 11106, 11142, 11196, 11268, 11358, 11412, 11538, 11574, 11646, 11754, 11862, 11898, 11916, 12114, 12132, 12186, 12294, 12438, 12492, 12564, 12618, 12708, 12762, 12924, 12942, 13086, 13194, 13212, 13302, 13374, 13428, 13518, 13626, 13644, 13698, 13788, 13842, 13914, 14004, 14166, 14184, 14292, 14328, 14346, 14436, 14562, 14598, 14724, 14778, 14814, 14886, 14922, 15084, 15102, 15156, 15192, 15354, 15426, 15462, 15516, 15534, 15588, 15786, 15804, 15858, 15894, 15948, 15966, 16056, 16164, 16326, 16344, 16398, 16452, 16488, 16542, 16596, 16668, 16722, 16776, 16812, 16866, 16938, 17046, 17154, 17208, 17244, 17352, 17406, 17478, 17532, 17586, 17676, 17694, 17838, 17946, 17964, 18072, 18108, 18162, 18234, 18324, 18342, 18378, 18504, 18558, 18594, 18702, 18756, 18828, 18882, 18918, 18936, 19098, 19134, 19242, 19368, 19476, 19512, 19566, 19638, 19674, 19692, 19746, 19854, 19944, 19962, 20052, 20106, 20214, 20232, 20268, 20322, 20376, 20484, 20556, 20718, 20754, 20772, 20934, 21078, 21096, 21132, 21258, 21348, 21366, 21474, 21564, 21618, 21636, 21834, 21852, 21906, 22014, 22068, 22104, 22122, 22158, 22212, 22266, 22284, 22300, 22392, 22482, 22536, 22662, 22700, 22716, 22824, 22900, 22986, 23022, 23076, 23094, 23148, 23202, 23238, 23292, 23300, 23346, 23418, 23454, 23508, 23526, 23724, 23742, 23778, 23796, 23832, 23886, 23900, 24100, 24228, 24264, 24372, 24498, 24588, 24606, 24714, 24858, 24876, 24984, 25100, 25128, 25182, 25236, 25362, 25416, 25524, 25614, 25686, 25700, 25722, 25794, 25848, 25884, 25902, 26046, 26118, 26154, 26172, 26262, 26300, 26388, 26424, 26478, 26604, 26658, 26694, 26748, 26766, 26802, 26856, 26874, 26900, 26982, 27036, 27100, 27198, 27252, 27288, 27396, 27414, 27558, 27576, 27684, 27700, 27774, 27828, 27882, 27954, 28008, 28062, 28100, 28206, 28278, 28300, 28332, 28422, 28494, 28584, 28692, 28746, 28818, 28872, 28926, 28962, 29034, 29124, 29142, 29178, 29196, 29286, 29300, 29448, 29466, 29556, 29628, 29772, 29826, 29844, 29934, 30006, 30042, 30168, 30204, 30312, 30474, 30546, 30582, 30700, 30708, 30762, 30852, 30924, 30978, 31014, 31032, 31068, 31100, 31176, 31194, 31300, 31338, 31446, 31554, 31572, 31608, 31662, 31700, 31716, 31788, 31896, 31932, 31986, 32094, 32166, 32202, 32328, 32418, 32598, 32652, 32796, 32814, 32904, 32958, 33084, 33100, 33192, 33246, 33336, 33444, 33498, 33606, 33624, 33678, 33700, 33714, 33732, 33786, 33822, 33876, 34002, 34092, 34218, 34308, 34326, 34434, 34488, 34700, 34758, 34794, 34812, 34900, 34956, 35064, 35082, 35118, 35172, 35300, 35352, 35388, 35514, 35622, 35676, 35766, 35874, 35892, 35900, 35928, 35946, 35982, 36054, 36198, 36216, 36306, 36324, 36468, 36486, 36522, 36648, 36684, 36700, 36702, 36756, 36954, 37116, 37134, 37188, 37242, 37300, 37404, 37458, 37494, 37512, 37566, 37602, 37656, 37764, 37782, 37836, 37900, 37998, 38034, 38196, 38268, 38300, 38322, 38358, 38466, 38484, 38538, 38574, 38754, 38898, 38900, 38952, 39132, 39222, 39276, 39348, 39384, 39492, 39654, 39700, 39708, 39726, 39834, 39924, 39978]

There are 146 abundant numbers with an \( \textbf{odd}\) sum of divisors. These are up to 40000:

[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]

The total number is 718

Note how I've marked 11025 in red. This is because it is the only odd number with an odd number of divisors. We have:$$11025 = 3^2 \times 5^2 \times 7^2 = 105^2 \text{ with 27 divisors}$$The next such number is 99225 where:$$99225=3^4 \times 5^2 \times 7^2 = 315^2 \text{ with 243 divisors}$$

Let's not confuse oddness and evenness of the divisor sums with the oddness and evenness of the numbers themselves. The majority of Zumkeller numbers are even. To quote from an earlier blog of mine:

In the range up to 100,000 there are 24362 even Zumkeller numbers comprising 24.362% of the range. However, there are only 208 odd Zumkeller numbers in that range, comprising 0.208%.

One of a Kind?

Consider all the natural numbers up to one million and ask the following question:

What numbers are comprised of the same set of digits as comprise its prime factors? It turns out that 132 is the first such number:$$ \begin{align} 132 &= 2^2 \times 3 \times 11 \\ \{1, 2, 3 \} &= \{1, 2, 3 \} \end{align} $$The second such number is 312:$$ \begin{align} 312 &= 2^3 \times 3 \times 13 \\ \{1, 2, 3 \} &= \{1, 2, 3\} \end{align} $$The third such number is 735:$$ \begin{align} 735 &= 3 \times 5 \times 7^2 \\  \{3, 5, 7\} &= \{3, 5, 7\} \end{align} $$It can be noted that 735 is unique (at least up to one million) in that its factors are single digits and thus none need to be broken down further.

These types of numbers comprise OEIS A035141:

A035141

Composite numbers \(k\) such that digits in \(k\) and in juxtaposition of prime factors of \(k \) are the same (apart from multiplicity).

There are 96 such numbers in the range up to 40,000:

132, 312, 735, 1255, 1377, 1775, 1972, 3792, 4371, 4773, 5192, 6769, 7112, 7236, 7371, 7539, 9321, 11009, 11099, 11132, 11163, 11232, 11255, 11375, 11913, 12176, 12326, 12595, 12955, 13092, 13175, 13312, 13377, 13491, 13755, 14842, 15033, 15303, 15317, 15532, 16332, 17272, 17276, 17343, 17482, 17973, 17975, 19075, 19276, 20530, 21345, 21372, 22413, 22714, 23535, 24338, 25030, 25105, 27232, 27393, 27944, 31007, 31317, 31419, 31479, 31503, 31592, 31722, 31977, 32024, 32104, 32145, 32612, 32973, 33011, 33327, 33781, 33925, 34112, 34997, 35213, 35262, 35722, 36882, 37115, 37127, 37317, 37359, 37522, 37662, 37741, 37791, 37921, 38385, 39172, 39795

However, it can be restated that 735 is very special, if not unique, because its prime factors (without multiplicity) of 3, 5 and 7 are the same as the digits of the number.

Primitive Sets

The Erdős primitive set conjecture is that the following summation:$$ \sum _{n \, \in A} \frac {1}{n\log{n}}$$where A is any primitive set (a set where no member of the set divides another member) attains its maximum at the set of primes numbers. It was proved by Jared Duker Lichtman (pictured above) in 2022. I was informed of this via a YouTube video first released in 2022. Here is a link to the academic paper by Lichtman. The constant turns out to be about 1.6366 ... and summations of the form shown above can be no larger than this. Thus we have:$$ \sum _{p \, \in P} \frac {1}{p\log{p}} \approx 1.6366$$where P is the set of primes and \(p\) is any prime number.

Let's take the summation of the elements of the set S of semiprimes. These elements form a primitive set. Let's suppose each semiprime can be represented by its prime factors \(p\) and \(q\) where \(p \leq q\). We then have:$$ \sum _{pq \, \in S} \frac {1}{pq\log {pq}} \approx 1.1448 \dots$$We can continue this process and consider the primitive set containing all numbers with three not necessarily distinct prime factors and so on. In each case, the sum converges to a constant which can be designated as \(f_k\) where \(k\) represents the number of prime factors. Thus we've seen that \(f_1 \approx 1.6366\) and \(f_2 \approx 1.1448\). In general we write:$$f_k=\sum  \frac {1}{n\log{n}}\\ \text{where } n \text{ has } k \text{ prime factors}$$The long term behaviour of \(f_k\) is shown in Figure 1 where the term "fingerprint numbers" is used to identify these types of numbers:

Figure 1: screenshot from video

While the values of \(f_k\) initially decrease and drop below 1 for \(f_3\), it can be seen that the values bottom out around \(f_5\) and \(f_6\) and then increase slowly as they approach a value of 1 asymptotically from below. This behaviour can be seen more clearly in Figure 2.

Figure 2: screenshot from video

The term "fingerprint numbers" is an informal term used in the referenced videos and is not used formally in mathematical circles. There are many different types of primitive sets but they all have the property that no member of the set divides another member.