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Jan
26
comment Trying to understand a step in Ramanujan's Proof of Bertrand's Postulate regarding the gamma function
Yes unfortunate about the notation. It is not the digamma function. Ramanujan's version of this proof is clearer IMO (I looked at the one you are working from). The main definition that is omitted from your version is $\log[x]! = \psi(x)+\psi(x/2)+\psi(x/3)+...$ I have to run but glad to help if someone else doesn't beat me to it.
Jan
25
comment What is the inverse of the divisor sum function $\sigma $?
I wonder if the title shouldn't be changed. You are looking for something very specific and neither the title nor the question make it clear exactly what that is.
Jan
24
comment Numbers that are divisible by the number of primes smaller than them
@PeterWoolfitt: $175197/\pi(175197) = 11.$
Jan
24
comment Numbers that are divisible by the number of primes smaller than them
Your idea is true through k=14 (which causes my computer to smoke).
Jan
24
comment Numbers that are divisible by the number of primes smaller than them
How far did you check, out of curiosity?
Jan
24
comment Numbers that are divisible by the number of primes smaller than them
k = 11 only occurs once so the "intuition" seems shaky.
Jan
24
comment Paul Erdős showed a simple estimate for $\pi(x) \ge \frac{1}{2}\log_2 x$; is it possible to tweak his argument to improve the estimate?
Considering the research that has gone into approximations of $\pi(x)$ it is no surprise that an approximation $\hat{\pi}(x)$ and a factor $(1-\epsilon)$ give a lower bound $\hat{\pi}(1-\epsilon)$ that is better than something derived for another purpose. Even Dusart's weaker bounds (Wiki) are much better than the one here. Nice idea though.
Jan
23
comment Question about Paul Erdős’ proof on the infinitude of primes
Oh--thanks for some reason I didn't see any message notice for the (probably another came up at same time). Appreciate it.
Jan
23
comment What is the inverse of the divisor sum function $\sigma $?
@ndroock1: is the object still to show that $\sigma*\sigma^{-1}= I?$
Jan
23
comment What is the inverse of the divisor sum function $\sigma $?
@ErickWong: that's theorem 2.20 (in the one-line proof).
Jan
23
comment What is the inverse of the divisor sum function $\sigma $?
Is Apostol's Theorem 2.20 any better?
Jan
23
comment Question about Paul Erdős’ proof on the infinitude of primes
Would it be possible to give the page reference for the passage in Havil since it is not in the index?
Jan
23
comment Finding the largest factorial with only three distinct decimal digits
The tags for this question are not really right if the question is about the code.
Jan
23
comment Consecutive numbers that share the same sum of prime factors
Mathematica is very good for this purpose. It lists the first 200000 values of f in about 10 seconds. With a little extra effort you could get it to find pairs $f(n+1)=f(n).$
Jan
22
comment Consecutive numbers that share the same sum of prime factors
How far is "not very far?"
Jan
22
comment Consecutive numbers that share the same sum of prime factors
Related to Catalan's conjecture how? How far have you searched?
Jan
21
comment Applying iterated function on the sum of the squares of the prime factors of $30$
Actually that last one is tricky. Not sure I agree. What I would say is that if $n$ is sufficiently large (50K or so) f(n) will be greater than 30. This lemma if true only deals with first iterates. So suppose $n = 50000.$ Then $f(n)$ will be greater than 30. Suppose $f(n)$ is much less than 50000. Then I think $f(f(n)$ might be less than 30.
Jan
21
comment Applying iterated function on the sum of the squares of the prime factors of $30$
I think it is possible to show that $\frac{2\ln n}{\ln 2}< f(n)\leq n^2.$ Then for example we need look no farther than $n = 50000$ to find $f(n) = 30.$ Not very useful.
Jan
21
revised How did Gauss discover the prime number theorem?
added 7 characters in body
Jan
20
answered How did Gauss discover the prime number theorem?