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Apr
15
comment Sum of products of $(1 - 1/p)$
@user254665 Thanks, I know it diverges; I want to know the rate at which it does so: probably something between $O(\log \log n)$ (which is the asymptotics of the sum of reciprocals of primes) and $O(n)$.
Apr
15
comment Sum of products of $(1 - 1/p)$
@Did Thanks, I have edited the question. I see the ambiguity now! Thanks user1952009 too!
Apr
15
comment Sum of products of $(1 - 1/p)$
@user1952009 I mean $S_N = \sum_{m=1}^{\pi(N)} \prod_{k=1}^m (1-\frac{1}{p_k})$
Apr
15
comment Sum of products of $(1 - 1/p)$
@user1952009 All the terms of the series are different; there is no weighting. I'm trying to understand the reason for the confusion... how could I make it clearer?
Apr
15
comment Sum of products of $(1 - 1/p)$
@Did I'm afraid I don't understand the difference -- I put the final term of $S_n$ only to indicate that we stop at the largest prime $\le n$; otherwise the typical expression is the same, right?
Mar
27
comment Time complexity of a modulo operation
There is a bug in the above: The termination condition should not be $s < q$, but running out of digits of $p$.
Mar
18
comment 2015-related question: why are Lucas-Carmichael numbers named after Lucas?
I haven't understood or verified it, but according to mersenneforum.org/showthread.php?t=3359, a Lucas–Carmichael number "is a Lucas pseudoprime to all Lucas sequences". There are also other definitions of Carmichael–Lucas numbers in the literature, so I don't think this terminology is very solid.
Feb
18
comment Finding where the tail starts for a probability distribution, from its generating function
Thank you for this excellent and very clear answer and for the link to the Goulden-Jackson cluster method; I will try to learn it.
Feb
16
comment Finding where the tail starts for a probability distribution, from its generating function
@Kirill: Thanks for the explanation; makes sense now. One question (drifting from my original question) is whether we can actually write down bounds for the tail sum (over coefficients $n > l$), beyond just saying that it's asymptotically $C\left(\frac{1}{\alpha}\right)^l$: can we bound the error, for a fixed finite $l$?
Feb
15
comment Finding where the tail starts for a probability distribution, from its generating function
@Kirill Can you elaborate on how you found $C$? Not sure how to get the partial fractions…
Feb
15
comment Finding where the tail starts for a probability distribution, from its generating function
Very nice, thank you! I knew that the $p_n$s would asymptotically be $A\left(\frac{1}{\alpha}\right)^n$, but thought that as this would only become a good approximation "eventually"; it wasn't useful as the specific details of the early terms would matter a lot. I am glad to have been wrong. :-)
Feb
10
comment Finding where the tail starts for a probability distribution, from its generating function
@Marcel: Thanks, can that be used to compute the bounds quickly? A priori it doesn't seem like doing the integration for every $m$ will be any faster than just computing the coefficients of the probability generating function directly (using symbolic algebra methods).
Jan
20
comment Mathematics understood through poems?
Updating the broken links: maa.org/press/periodicals/convergence/… maa.org/press/periodicals/convergence/… etc.
Dec
15
comment Expected Number of Coin Tosses to Get Five Consecutive Heads
I found an answer using martingales here: quora.com/… but I'm curious if there is a generating functions way (also about the distribution, say variance or number of trials until 90% probability of seeing what we want).
Dec
15
comment Expected Number of Coin Tosses to Get Five Consecutive Heads
Do you know how to find the distribution (or expectation and variance) for the number of tosses until either 5 consecutive heads or 5 consecutive tails? (Or 5 consecutive equal results from rolling dice.) Is there a question on math.se about this?
Nov
13
comment $e^{e^{e^{79}}}$ and ultrafinitism
@FanZheng Bases don't have to be integers; see Wikipedia articles on Non-integer representation and base-φ. For the rest, I've already explained the issue in my previous comments; I don't know which part is unclear. You might also want to look at this question which is about the "every base is base 10" joke.
Sep
11
comment Is this Batman equation for real?
I just found out that the curve was devised by J. Mathew Register (then "teaching at a few art schools throughout the greater Sacramento area", now "a full time professor over at American River College"): quora.com/Who-wrote-the-Batman-equation/answer/…
Jul
27
comment Recurrence relation with generating function problem
I agree; that's part of why it's less simple in this case.
Jul
22
comment Fast way to get a position of combination (without repetitions)
@van E.g. on SO: stackoverflow.com/questions/3143142/… stackoverflow.com/questions/5307222/… (and many others). On this site: math.stackexchange.com/questions/349924/… (and many others). On Wikipedia: en.wikipedia.org/wiki/Combinatorial_number_system en.wikipedia.org/wiki/Factorial_number_system Terms to search for are "(ranking|indexing) combinations", also "combinadics", and names of those WP articles.
Jul
22
comment Simple string permutations question
The answer is correct, but if you're writing it up for homework / an exam etc., you should add that $\binom{5}{2}$ is the number of ways to allocate the positions of the vowels in the sequence.