Weird factorial and anti-primes

Number Theory Level 3

Anti-primes (or highly composite numbers) are numbers that have more factors than all the positive numbers before it

Example: 2 is a highly composite number because it has more factors than 1 (2 is a prime, has two divisors,1 has only 1 factor and that is itself)

Using this logic, 1!,2!,3!,4!,5!,6!,7! are highly composite numbers

So the question is: Is 8! a highly composite number?

Note

No Yes

1 solution

Mr. India
Mar 16, 2019

No. of factors of a number $N=a^p.b^q.c^r....(a,b,c$ are primes) is

$(p+1)(q+1)(r+1).....$

$7!=1.2.3.4.5.6.7=2^4.3^2.5^1.7^1$

No. Of Factors of $7!=5.3.2.2=60$

$8!=7!×8=2^7.3^2.5^1.7^1$

No. Of factors $=8.3.2.2=96$

$8!=2^7.3^2.5^1.7^1>27720(2^3.3^2.5^1.7^1.11^1)$

No. of factors of above no. $=4.3.2.2.2=96$

So, they both have same number of factors and therefore, $\boxed{27720}$ is a super composite number and not $8!$

@Gia Hoàng Phạm , from where did you get to know about these numbers??

Mr. India - 2 years, 2 months ago