Why are there more numbers with 4 divisors than 2?

Answer 1

The only numbers with a pair of divisors would be prime numbers which will have factors of 1 and themselves.

Numbers with 4 divisors would be these numbers multiplied by a prime number. For example if you had the number 2 (2 factors of 1 and 2), you could multiply it by 3 to get 6. Six has factors of 1, 2, 3 and 6. But you could also multiply it by 5 (1, 2, 5, 10)... or 7 (1, 2, 7, 14).. or 11 (1, 2, 11, 22), etc.

Now technically there are an infinite number of numbers in both sets but relative to each other, there are generally more numbers in a given interval that have 4 divisors rather than just 2 divisors.

Answer 2

I agree with puzzling that the numbers with only two divisors are primes, those with four divisors are then composites, but are you intending to disallow 1 as a divisor? In any case, it is not true that there are more numbers with four divisors than there are with two. I know it is counter-intuitive, but we are in the realm of transfinite arithmetic. If N(2) is the set of all numbers with exactly two divisors and N(4) is the set of all numbers with exactly four divisors, then N(2) and N(4) "have the same cardinality".

Answer 3

First of all, that's not true overall. Both counts are infinite.

Second, it's not true in every interval. For example, in the interval [5,7] it's false.

Third, there are two cases where integers have exactly 4 divisors. One is a product of distinct primes. The other is the cube of a prime.

Answer 4

You ask very hard questions?????