This article was inspired by a vlog from Numberphile, on the discussion of “5040: an anti-prime number”, or some title like that.

A contributor to the OEIS named `Jean-François Alcover` came up with a short bit of Mathematica code that I modified slightly:

Reap[ For[ record = 0; n = 1, n <= 110880, n = If[n < 60, n + 1, n + 60], tau = DivisorSigma[0, n]; If[tau > record, record = tau; Print[n, "\t\t", tau]; Sow[tau]]]][[2, 1]]

This generates a list of a set of numbers with an unusually high amount of factors called “highly composite numbers” up to 110,880. The second column of the output are the number of factors.

1 1 2 2 4 3 6 4 12 6 24 8 36 9 48 10 60 12 120 16 180 18 240 20 360 24 720 30 840 32 1260 36 1680 40 2520 48 5040 60 7560 64 10080 72 15120 80 20160 84 25200 90 27720 96 45360 100 50400 108 55440 120 83160 128 110880 144

For a number like 110,880, there is no number before it that has more than 144 factors.

Highly composite numbers (HCNs) are loosely defined as a natural number which has more factors than any others that came before it. 12 is such a number, with 6 factors, as is 6 itself with 4. The number 5040 has 60 factors, and is also considered highly composite.

5040=2^{4}×3^{2}×5×7

This works out to 60, because with 2^{4}, for example, we get the factors 2, 4, 8, and 16. With 2^{4}×3^{2}, we get 2, 3, 4, 6, 8, 9, 16, 18, 36, 72, and 144, all which evenly divide 5040. The total number of factors including 1 and 5040 itself can be had from adding 1 to each exponent and multiplying: (4+1)(2+1)(1+1)(1+1)=5×3×2×2=60.

Initially, facotorization of HCNs was done in Maple using the “ifactor()” command. But there is a publication circulating the Internet referring to a table created by Ramanujan that has these factors. A partial list of these are summarized in a table below. The top row headers are the prime numbers that can be the prime factors, from 2 to 17. The first column is the number to factorize. The numbers in the same columns below these prime numbers are the exponents on the primes, such as: 10,080=2^{5}×3^{2}×5^{1}×7^{1}. The last column are the total number of factors on these HCNs. So, by adding 1 to each exponent in the row and multiplying, we find that 10,080 has 6×3×2×2=72 factors.

###### NUMBER PATTERNS OBSERVED

As a number of factors (underneath the “# facotrs” column), We get overlapping patterns starting from 60. One of them would be the sequence: 120, 240, 360, 480, 600, and 720. But the lack of an 840 breaks that pattern. But then we get 960, then 1080 is skipped, but then we get 1200.

For numbers of factors that are powers of 2, it seems to go right off the end of the table and beyond: 64, 128, 256, 512, 1024, 2048, 4096, 8192, … . Before 5040, the pattern is completed, since 2 has 2 factors, 6 has 4 factors, 24 has 8 factors, 120 has 16 factors, and 840 has 32 factors. The HCN with 8192 factors is 3,212,537,328,000. We have to go beyond that to see if there is a number with 16,384 factors.

Multiples of 12 make their appearance as numbers of factors: 12, 24, 36, 48, 60 (which are the numbers of factors of 5040), 72, 84, 96, 108, 120, but a lack of a 132 breaks that pattern. But then we see: 144, 288, 432, 576, 720, 864, 1008, 1152, and the pattern ends with the lack of a 1296.

We also observe short runs of numbers of factors in the sequence 100, 200, 400, 800, until we reach the end of this table. But the pattern continues with the number 2,095,133,040, which has 1600 factors. Then, 3200 is skipped.

There are also multiples of 200: 200, 400, 600, 800, but the lack of a 1000 breaks that pattern. But when seen as multiples of 400, we get: 400, 800, 1200, 1600, but then 2000 is skipped.

There are also peculiarities in the HCNs themselves. Going from 5040 to as high as 41,902,660,800, only 4 of the 60 HCNs were not multiples of 5040. The rest had the remainder 2520, which is one-half of 5040.

Also beginning from the HCN 720,720, we observe a run of numbers containing 3-digit repeats: 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640, 10810800, and 14414400.

Number 2 3 5 7 11 13 17 # of factors ----------------------------------------------------------------------- 5040 4 2 1 1 60 7560 3 3 1 1 64 10080 5 2 1 1 72 15120 4 3 1 1 80 20160 6 2 1 1 84 25200 4 2 2 1 90 27720 3 2 1 1 1 96 45360 4 4 1 1 100 50400 5 2 2 1 108 55440 4 2 1 1 1 120 83160 3 3 1 1 1 128 110880 5 2 1 1 1 144 166320 4 3 1 1 1 160 221760 6 2 1 1 1 168 332640 5 3 1 1 1 192 498960 4 4 1 1 1 200 554400 5 2 2 1 1 216 665280 6 3 1 1 1 224 720720 4 2 1 1 1 1 240 1081080 3 3 1 1 1 1 256 1441440 5 2 1 1 1 1 288 2162160 4 3 1 1 1 1 320 2882880 6 2 1 1 1 1 336 3603600 4 2 2 1 1 1 360 4324320 5 3 1 1 1 1 384 6486480 4 4 1 1 1 1 400 7207200 5 2 2 1 1 1 432 8648640 6 3 1 1 1 1 448 10810800 4 3 2 1 1 1 480 14414400 6 2 2 1 1 1 504 17297280 7 3 1 1 1 1 512 21621600 5 3 2 1 1 1 576 32432400 4 4 2 1 1 1 600 61261200 4 2 2 1 1 1 1 720 73513440 5 3 1 1 1 1 1 768 110270160 4 4 1 1 1 1 1 800 122522400 5 2 2 1 1 1 1 864 147026880 6 3 1 1 1 1 1 896 183783600 4 3 2 1 1 1 1 960 245044800 6 2 2 1 1 1 1 1008 294053760 7 3 1 1 1 1 1 1024 367567200 5 3 2 1 1 1 1 1152 551350800 4 4 2 1 1 1 1 1200

After that run, we see a 4-digit overlapping repeat. The digits of the HCN 17297280 could be thought of as an overlap of 1728 and 1728 to make 1729728 as part of that number. The 3-digit run continues with: 21621600, 32432400, 61261200, and after that the pattern is broken.