Is $\ 7!=5040\ $ the largest highly composite factorial?

A highly composite number is a natural number $\ n\ge 1\ $ that has more divisors than any smaller natural number $\ m\ge 1$.

Checking the $\ 1000\ $ entries in the OEIS-sequence, I noticed that only the following factorials are highly composite :

$$[1, 2, 6, 24, 120, 720, 5040]$$

Is it known whether $\ 7!=5040\ $ is the largest highly composite factorial ?

The factorials have some conditions necessary for a number to be highly composite :

• They contain the first $k$ prime numbers as prime factors.
• The exponents are non-increasing.
• The exponent corresponding to the largest prime factor is $\ 1$.

So, it is not obvious whether the above list is complete.

3 Answers

Suppose $n \ge 20$. Trivially, $n!$ is then divisible by $16$, so $\dfrac{13}{16}n!$ is an integer smaller than $n!$.

Let the prime factorization of $n!$ be $n! = 2^{e_1}3^{e_2}\cdots11^{e_5}13^{e_6}17^{e_7}\cdots p_r^{e_r}$.

Then the prime factorization of $\dfrac{13}{16}n!$ is $\dfrac{13}{16}n! = 2^{e_1-4}3^{e_2} \cdots11^{e_5}13^{e_6+1}17^{e_6}\cdots p_r^{e_r}$.

Then, the number of divisors of $n!$ is $\sigma_0(n!) = (e_1+1)(e_6+1)\displaystyle\prod_{k \neq 1,6}(e_k+1)$ and the number of divisors of $\dfrac{13}{16}n!$ is $\sigma_0(\dfrac{13}{16}n!) = (e_1-3)(e_6+2)\displaystyle\prod_{k \neq 1,6}(e_k+1)$.

Hence, $\sigma_0(\dfrac{13}{16}n!) > \sigma_0(n!)$ iff $(e_1-3)(e_6+2) > (e_1+1)(e_6+1)$ iff $e_1 > 4e_6+7$.

Using the well known formula for the largest power of a prime dividing a factorial, we have

$e_1 = \displaystyle\sum_{k = 1}^{\left\lfloor \log_2 n\right\rfloor}\left\lfloor \dfrac{n}{2^k} \right\rfloor > \displaystyle\sum_{k = 1}^{\left\lfloor \log_2 n\right\rfloor} \left(\dfrac{n}{2^k} - 1\right) \ge n - \dfrac{n}{2^{\left\lfloor \log_2 n\right\rfloor}} - \left\lfloor \log_2 n\right\rfloor \ge n - 2 - \log_2 n$.

Similarly, $e_6 = \displaystyle\sum_{k = 1}^{\left\lfloor \log_{13} n\right\rfloor}\left\lfloor \dfrac{n}{13^k} \right\rfloor \le \displaystyle\sum_{k = 1}^{\left\lfloor \log_{13} n\right\rfloor}\dfrac{n}{13^k} \le \dfrac{n}{12}$. Hence, $4e_6+7 \le \dfrac{n}{3}+7$.

I'll leave it as an exercise to show that $\dfrac{n}{3}+7 < n - 2 - \log_2 n$ holds for all integers $n \ge 20$.

Therefore, for all integers $n \ge 20$, we have $4e_6+7 \le \dfrac{n}{3}+7 < n - 2 - \log_2 n < e_1$, and thus, $\sigma_0(\dfrac{13}{16}n!) > \sigma_0(n!)$.

So, $n!$ is not highly composite for $n \ge 20$. Now, it remains to check if any factorials between $7!$ and $20!$ are highly composite.

EDIT: From the list of the first 1000 highly composite numbers that OP mentioned, it appears that the $149$-th largest highly composite number is $\approx 1.49 \times 10^{17}$, while $19! \approx 1.22 \times 10^{17}$. None of the numbers $8!, 9!, \ldots, 19!$ appear in that list, so $7!$ is indeed the largest highly composite factorial.

Yes, it is known. If you look at the number of divisors function, a number $n$ with prime factorization $p_1^{a_1}p_2^{a_2}p_3^{a_3}\ldots$ has $\sigma_0(n)=(a_1+1)(a_2+1)(a_3+1)\ldots$ divisors. The factorials have too many small factors. Every time you multiply by a prime you double the number of divisors, but when you increase the power of $2$ from $n$ to $n+1$ you only multiply the number of divisors by $\frac {n+2}{n+1}=1+\frac 1{n+1}$. The greedy algorithm for finding highly composite numbers would look at $\log \sigma_0(n)$. If you multiply $n$ by a prime $p$ that is currently in the factorization of $n$ as $p^a$ you add $\log 1+\frac 1{a+1}$ to the log of the number of factors and you add $\log p$ to the log of $n$, so the benefit/cost ratio of each increase is $\frac {\log 1+\frac 1{a+1}}{\log p}$ Search over the primes, find the best, and that is your next multiplier. When you look at OEIS A002182 not every entry is a multiple of the previous one, but you see there are too many small factors among the factorials. $n!$ has about $\frac n2$ factors of $2$, for example. To really justify this you need to use the prime number theorem to show how large the next prime will be, then show you want to use it before $n!$ will. You can find a finite limit of factorials that are candidates, then check them.

What you are missing is the ratio of exponents. For factorials, by the theorem of Legendre, the exponent of $p$ is proportional to $1/p.$ I suppose it is slightly more accurate to say proportional to $1 /(p-1).$

The superior highly composite numbers have exponents for each prime proportional to $1/ \log p.$ Guy Robin made a method for interpolating between consecutive SHC numbers to construct all highly composite numbers between. The point is that highly composite numbers are, in the long run, similar to the least common multiple of the numbers from $1$ to some $m.$

It might be a good deal of work to show that $5040$ is the last factorial that works, but it is clear that there are only finitely many such, and explicit bounds could be constructed. From the output below, you can see how much smaller an SHC number can be, yet have more divisors than the factorial under consideration. The logarithms are base ten.

jagy@phobeusjunior:~$ ./factorial 2! 2 = 2 div 2 log 0.30103 3! 6 = 2 3 div 4 log 0.778151 4! 24 = 2^3 3 div 8 log 1.38021 5! 120 = 2^3 3 5 div 16 log 2.07918 6! 720 = 2^4 3^2 5 div 30 log 2.85733 7! 5040 = 2^4 3^2 5 7 div 60 log 3.70243 8! 40320 = 2^7 3^2 5 7 div 96 log 4.60552 9! 362880 = 2^7 3^4 5 7 div 160 log 5.55976 10! 3628800 = 2^8 3^4 5^2 7 div 270 log 6.55976 11! 39916800 = 2^8 3^4 5^2 7 11 div 540 log 7.60116 12! 479001600 = 2^10 3^5 5^2 7 11 div 792 log 8.68034 13! 6227020800 = 2^10 3^5 5^2 7 11 13 div 1584 log 9.79428 14! 87178291200 = 2^11 3^5 5^2 7^2 11 13 div 2592 log 10.9404 15! 1307674368000 = 2^11 3^6 5^3 7^2 11 13 div 4032 log 12.1165 16! 20922789888000 = 2^15 3^6 5^3 7^2 11 13 div 5376 log 13.3206 17! 355687428096000 = 2^15 3^6 5^3 7^2 11 13 17 div 10752 log 14.5511 18! 6402373705728000 = 2^16 3^8 5^3 7^2 11 13 17 div 14688 log 15.8063 19! 121645100408832000 = 2^16 3^8 5^3 7^2 11 13 17 19 div 29376 log 17.0851 20! 2432902008176640000 = 2^18 3^8 5^4 7^2 11 13 17 19 div 41040 log 18.3861 21! 51090942171709440000 = 2^18 3^9 5^4 7^3 11 13 17 19 div 60800 log 19.7083 22! 1124000727777607680000 = 2^19 3^9 5^4 7^3 11^2 13 17 19 div 96000 log 21.0508 23! 25852016738884976640000 = 2^19 3^9 5^4 7^3 11^2 13 17 19 23 div 192000 log 22.4125 24! 620448401733239439360000 = 2^22 3^10 5^4 7^3 11^2 13 17 19 23 div 242880 log 23.7927 25! 15511210043330985984000000 = 2^22 3^10 5^6 7^3 11^2 13 17 19 23 div 340032 log 25.1906 jagy@phobeusjunior:~$ ./Superior_Highly_Composite_read 2 = 2 div 2 log 0.30103 6 = 2 3 div 4 log 0.778151 12 = 2^2 3 div 6 log 1.07918 60 = 2^2 3 5 div 12 log 1.77815 120 = 2^3 3 5 div 16 log 2.07918 360 = 2^3 3^2 5 div 24 log 2.5563 2520 = 2^3 3^2 5 7 div 48 log 3.4014 5040 = 2^4 3^2 5 7 div 60 log 3.70243 55440 = 2^4 3^2 5 7 11 div 120 log 4.74382 720720 = 2^4 3^2 5 7 11 13 div 240 log 5.85777 1441440 = 2^5 3^2 5 7 11 13 div 288 log 6.1588 4324320 = 2^5 3^3 5 7 11 13 div 384 log 6.63592 21621600 = 2^5 3^3 5^2 7 11 13 div 576 log 7.33489 367567200 = 2^5 3^3 5^2 7 11 13 17 div 1152 log 8.56534 6983776800 = 2^5 3^3 5^2 7 11 13 17 19 div 2304 log 9.84409 13967553600 = 2^6 3^3 5^2 7 11 13 17 19 div 2688 log 10.1451 321253732800 = 2^6 3^3 5^2 7 11 13 17 19 23 div 5376 log 11.5068 2248776129600 = 2^6 3^3 5^2 7^2 11 13 17 19 23 div 8064 log 12.3519 65214507758400 = 2^6 3^3 5^2 7^2 11 13 17 19 23 29 div 16128 log 13.8143 195643523275200 = 2^6 3^4 5^2 7^2 11 13 17 19 23 29 div 20160 log 14.2915 6064949221531200 = 2^6 3^4 5^2 7^2 11 13 17 19 23 29 31 div 40320 log 15.7828 12129898443062400 = 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 div 46080 log 16.0839 448806242393308800 = 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 div 92160 log 17.6521 18401055938125660800 = 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 div 184320 log 19.2648 791245405339403414400 = 2^7 3^4 5^2 7^2 11 13 17 19 23 29 31 37 41 43 div 368640 log 20.8983 

Made one just a little higher but without the factorizations, just the number of divisors and the logarithm base ten.

jagy@phobeusjunior:~$ ./factorial 2! 2 div 2 log 0.30103 3! 6 div 4 log 0.778151 4! 24 div 8 log 1.38021 5! 120 div 16 log 2.07918 6! 720 div 30 log 2.85733 7! 5040 div 60 log 3.70243 8! 40320 div 96 log 4.60552 9! 362880 div 160 log 5.55976 10! 3628800 div 270 log 6.55976 11! 39916800 div 540 log 7.60116 12! 479001600 div 792 log 8.68034 13! 6227020800 div 1584 log 9.79428 14! 87178291200 div 2592 log 10.9404 15! 1307674368000 div 4032 log 12.1165 16! 20922789888000 div 5376 log 13.3206 17! 355687428096000 div 10752 log 14.5511 18! 6402373705728000 div 14688 log 15.8063 19! 121645100408832000 div 29376 log 17.0851 20! 2432902008176640000 div 41040 log 18.3861 21! 51090942171709440000 div 60800 log 19.7083 22! 1124000727777607680000 div 96000 log 21.0508 23! 25852016738884976640000 div 192000 log 22.4125 24! 620448401733239439360000 div 242880 log 23.7927 25! 15511210043330985984000000 div 340032 log 25.1906 26! 403291461126605635584000000 div 532224 log 26.6056 27! 10888869450418352160768000000 div 677376 log 28.037 28! 304888344611713860501504000000 div 917280 log 29.4841 29! 8841761993739701954543616000000 div 1834560 log 30.9465 30! 265252859812191058636308480000000 div 2332800 log 32.4237 31! 8222838654177922817725562880000000 div 4665600 log 33.915 32! 263130836933693530167218012160000000 div 5529600 log 35.4202 jagy@phobeusjunior:~$ ./Superior_Highly_Composite_read 2 div 2 log 0.30103 6 div 4 log 0.778151 12 div 6 log 1.07918 60 div 12 log 1.77815 120 div 16 log 2.07918 360 div 24 log 2.5563 2520 div 48 log 3.4014 5040 div 60 log 3.70243 55440 div 120 log 4.74382 720720 div 240 log 5.85777 1441440 div 288 log 6.1588 4324320 div 384 log 6.63592 21621600 div 576 log 7.33489 367567200 div 1152 log 8.56534 6983776800 div 2304 log 9.84409 13967553600 div 2688 log 10.1451 321253732800 div 5376 log 11.5068 2248776129600 div 8064 log 12.3519 65214507758400 div 16128 log 13.8143 195643523275200 div 20160 log 14.2915 6064949221531200 div 40320 log 15.7828 12129898443062400 div 46080 log 16.0839 448806242393308800 div 92160 log 17.6521 18401055938125660800 div 184320 log 19.2648 791245405339403414400 div 368640 log 20.8983 37188534050951960476800 div 737280 log 22.5704 185942670254759802384000 div 983040 log 23.2694 9854961523502269526352000 div 1966080 log 24.9937 581442729886633902054768000 div 3932160 log 26.7645 1162885459773267804109536000 div 4423680 log 27.0655 12791740057505945845204896000 div 6635520 log 28.1069