**MORE FACTS ABOUT PRIME FACTORIZATION OF FACTORIALS.**

In the previous post, we introduced these functions, just as a small trick to calculate the limit we were looking for, but unlikely of what they seem to be, they are less artificial than expected.

**Legendre´s formula for the exponent of p in the prime factorization of n!:**

**Integer Approximation for the Legendre´s formula:**

The diference between one function and its approximation is the error function.

**Error Function for the Legendre´s formula:**

We can use to get another beautiful expression for the error function:

This function shows fractal behavior:

**Particular Values for **:

, References [1] and [2]

**More facts about Legendre´s **

gives also the number of 1’s in binary expansion of (or the sum of all its binary digits).

And if we extend the range of this formula, been any number not necessarily prime, then:

It gives the base repunits, and so for base :

It gives the Mersenne Numbers.

Amazingly, this uninteresting topic, at first sight, becomes a joint between: Repunits, Mersenne numbers, Factorials, primes, fractals, counting of digits…

Number Theory is it!

**References:**
[1]-A011371-n minus (number of 1’s in binary expansion of n). Also highest power of 2 dividing n!. The On-Line Encyclopedia of Integer Sequences!

[2]-A000120-1’s-counting sequence: number of 1’s in binary expansion of n (or the binary weight of n). The On-Line Encyclopedia of Integer Sequences!

[3]-Cooper, Topher and Weisstein, Eric W. “Digit Sum.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/DigitSum.html

**Archives:**
[a]-121809-Notes on Legendre´s Formula.nb

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