I made a table that shows the number of divisors for each number less than 500, and i think that there is a pattern, for example when there is a spike in the number of divisors the surrounding numbers will have only few divisors. But is there other patterns in the distribution of divisors? Why are the number of divisors distributed in this pattern.
Each bar means one divisor.
2 --
3 --
4 ---
5 --
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7 --
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Here is the python code that i used to generate the table.
b=1
d = 0
#generates a list of numbers.
while b<500:
b=b+1
x = 0.0
a = 0
#generates a list of numbers less than b.
while x<b:
x=x+1
#this will check for divisors.
if (b/x)-int(b/x) == 0.0:
a=a+1
print b, a*"-"
You're interested in the special case of the divisor function $\sigma_0(n)$, which returns the number of divisors of $n$. The numbers with more divisors than any smaller number are called highly composite and they are all representable as a product of primorials. Equivalently, the exponents of their prime factors must be constant or decreasing. I believe the portion on prime factor subsets at the end of the link on highly composite numbers is the cause of the patterns you're noticing.