All Questions
52 questions
8
votes
1
answer
672
views
Infinite series and sum of two squares
Consider the following infinite sequence $a(n)$ generated by
$$\sum_{n\geq0} a(n)q^n
=\frac{\sum_{k\geq0}F(2k+1)q^{\binom{k+1}2}}{\sum_{k\geq0} q^{\binom{k+1}2}}$$
where the $F(2k+1)$ are the odd ...
- 43.2k
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
71
votes
8
answers
12k
views
Possible new series for $\pi$
In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...
- 82.7k
16
votes
3
answers
4k
views
Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
15
votes
3
answers
1k
views
Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
- 2,689
10
votes
0
answers
350
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 2,836
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
4
votes
1
answer
308
views
3 divides coefficents of this $q$-series
Denote $\phi(q):=\prod_{j\geq1}(1-q^j)$ and let $\xi=e^{\frac{2\pi i}3}$ be a cube root of unity.
Define the sequence $u(n)$ by
$$\prod_{n\geq1}\prod_{s=1}^2(1-q^n\xi^{ns})(1-q^{2n}\xi^{ns})
=\sum_{n\...
- 43.2k
13
votes
1
answer
1k
views
Apéry's constant $\zeta(3)$ fastest convergent series
UPDATE Feb.02.2024
The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of ...
- 2,836
0
votes
1
answer
346
views
A combinatorial proof: where art thou?
Start by introducing the finite sums
$$A_n:=\sum_{m=1}^nq^m\prod_{j=1}^{m-1}(1-q^j) \qquad \text{and} \qquad
B_n:=\sum_{m=1}^nq^m\prod_{j=m+1}^n(1-q^j).$$
An algebraic proof is facile: Clearly, $A_1=...
- 43.2k
3
votes
1
answer
156
views
$q$-series and Stirling of the 1st kind
Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define
$$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}_b(q)=\sum_{m\...
- 43.2k
6
votes
3
answers
536
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.2k
1
vote
0
answers
158
views
Hankel transform of certain $\pm1$ sequences
The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$
where $s_2(k)$ is ...
- 43.2k
1
vote
1
answer
344
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k
5
votes
3
answers
2k
views
How many digits of $\sqrt{2}$ are known to date?
How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring ...
- 3,098
1
vote
0
answers
87
views
Doubly log-concave or doubly log-convex
Suppose $(a_k)_{k\geq0}$ is a sequence of real numbers. Consider the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$.
We say $(a_k)_k$ is log-concave (resp. log-convex) provided $\mathcal{L}a_k\geq0$ (...
- 43.2k
5
votes
1
answer
386
views
Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$?
Let $p \equiv 1 \pmod 4$ be a prime and $E_n$ denote the $n$-th Euler number. While investigating $E_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$)
\begin{align*}
\sum_{k =1}^{\frac{...
- 707
17
votes
1
answer
1k
views
Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
1
vote
1
answer
186
views
Connection between central factorial numbers and the Stern–Brocot tree
Consider the central factorial numbers of even indices formed by
$$U(n,k)=\frac1{(2k)!}\sum_{i=0}^{2k}(-1)^i\binom{2k}i(k-i)^{2n}.$$
Let $u(n,k):=U(n,k)\mod 2$. Define the triangle of numbers
$$A(r,j)=...
- 43.2k
12
votes
1
answer
406
views
Looking for a "clever" argument for a $q$-series identity
Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod_{k\geq1}...
- 43.2k
5
votes
3
answers
300
views
Closed formula for $(-1)$-Baxter sequences
The number of the so-called Baxter permutations of length $n$ is computed by
$$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a ...
- 43.2k
5
votes
1
answer
435
views
Limit on a certain double sum
While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense). Notice first this well-known fact in context
$$\sum_{n,m\geq1}\frac1{...
- 43.2k
7
votes
3
answers
933
views
In search of an alternative proof of a series expansion for $\log 2$
We all know the series expansion
$$\log 2=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n. \tag1$$
I also am able to use the method of Wilf-Zeilberger to the effect that
$$\log 2=3\sum_{n=1}^{\infty}\frac{(-1)^{...
- 43.2k
2
votes
1
answer
236
views
Divisibility of (finite) power sum of integers
Consider the power sum
$$S_a(b)=1^{2b}+2^{2b}+\cdots+(3a-2)^{2b}.$$
Let $\nu_3(x)$ denote the $3$-adic valuation of $x$.
QUESTION 1. (milder) Is this true?
$$\nu_3\left(\frac{S_a(b)}{S_a(1)}\right)=0....
- 43.2k
3
votes
0
answers
219
views
On partial sums of the Ramanujan sums
Let $n$ be a positive integer and $c_{m}(n)$ denote the $m^{th}$ Ramanujan sum at $n$. What is the best known estimate for $\sum_{m=1}^{N} c_{m}(n)$?
- 31
6
votes
1
answer
302
views
A 3rd formula for the central Delannoy numbers?
There are several in the literature proving the two alternative formulas for the (diagonal) Delannoy numbers; namely that
$$d_n=\sum_{k=0}^n\binom{n}k\binom{n+k}k=\sum_{k=0}^n\binom{n}k^22^k.$$
Each ...
- 43.2k
6
votes
2
answers
547
views
2-adic valuation of a certain binomial sum
Consider the sequence (of rational numbers) given by
$$a_n=\sum_{k=1}^n\binom{n}k\frac{k}{n+k}.$$
Let $s(n)$ be the sum of binary digits of $n$, i.e. the total number of $1$'s.
QUESTION. Is it true ...
- 43.2k
5
votes
1
answer
365
views
Power of $2$ dividing a specialized Mittag-Leffler polynomial
While studying the so-called Mittag-Leffler Polynomials, denoted $M_n(x)$, I was looking into the sequence $\frac1{n!}M_n(n)$ which takes the following form
$$a_n:=\sum_{k=1}^n\binom{n-1}{k-1}\binom{n}...
- 43.2k
1
vote
0
answers
108
views
Question related to sequence of recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$
Define radical of an integer Wiki
$$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$
Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{...
- 599
2
votes
2
answers
253
views
Approximation of a square with an irrational arithmetic progression
Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ be irrational. Does the arithmetic progression $(n\alpha )_{n\in\mathbb{N}}$ becomes arbitrarily close to squares?
More precisely, what can be said ...
- 3,574
2
votes
0
answers
212
views
show that sequence $\{(-1)^n\Upsilon_n\}$ is convergent and strictly decreasing
Edit: Few years ago, I have posted my claim on $\Upsilon$ function regarding prime number but recently I have observed, last observation turns false that's way, (by putting $\Upsilon$ value in ...
- 599
13
votes
3
answers
810
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
4
votes
1
answer
244
views
The function $\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^s}$: reference request or particular values at integers and abscissa of convergence
We denote for integers $m\geq 1$ the Möbius function as $\mu(m)$. With the help of a CAS, Wolfram Alpha online calculator, I was calculating certain values of $$\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{...
7
votes
1
answer
1k
views
Signed variant of the Flint Hills series
I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one:
$$
\sum_{n = 1}^\infty \...
- 17.2k
0
votes
1
answer
556
views
Order of magnitude of $\sum \frac{1}{\log^2{p}}$, or $\sum \frac{1}{\log^a{p}}$ for arbitrary power $a$ [closed]
In this MO question, it says that we have
$$ \sum_{p<n} \frac{1}{\log{p}} =\frac{n}{\log^2 n}+O\left(\frac{n\log\log n}{\log^3 n}\right).$$
where the sum is on all primes $p$, up to some max ...
- 4,897
5
votes
0
answers
161
views
A relation concerning the "sum of squares" counting function $r_2(n)$
This is a re-post from MSE as I did not get any response there.
Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...
- 2,249
3
votes
1
answer
441
views
What is the shortest length of an Egyptian fraction expansion for a given $p/q$?
An Egyptian fraction expansion is a sum of reciprocals of integers, for example:
$$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$
Every positive rational number $p/...
- 4,164
7
votes
1
answer
232
views
Is anything known about this class of series involving the divisor function?
I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it!
Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
- 661
3
votes
1
answer
631
views
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
- 128k
24
votes
2
answers
1k
views
If $x_{n+1}= \frac{nx_{n}^2+1}{n+1}$ then $x_{n}=1$
I asked this question at MSE, but I think it's more appropriated to MO.
Let $x_{n}$ be a sequence, such that $x_{n+1}= \dfrac{nx_{n}^2+1}{n+1}$ and $x_n>0$ for all $n$.
There is a positive ...
- 3,153
0
votes
0
answers
266
views
Completing a dyadic sum
Suppose I knew the behaviour of a given sum in every other interval, for example:
$$
\sum_{\substack{0\leq a \leq x\\ a\equiv 1 (k)}} \sum_{x/(a+k/2)< b \leq x/a} f(b) \sim g(x),
$$
for any $x>1$...
- 3,799
5
votes
1
answer
613
views
generating $q$-Catalan numbers
An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square_n$ denote all such paths....
- 43.2k
15
votes
4
answers
3k
views
No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
5
votes
1
answer
680
views
When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...
- 3,443
14
votes
1
answer
755
views
Generating function of the Thue-Morse sequence
Let $T$ be the generating function of the Thue-Morse sequence; thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 \...
- 23k
10
votes
1
answer
554
views
Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...
- 10.5k
7
votes
1
answer
283
views
On one class of Somos-like sequences
This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\{...
- 12.3k
1
vote
2
answers
534
views
How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$?
Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a
strongly additive function on positive integer number $m$, where $p$ is a prime number. Set
$${f_x}(p) = \left\{ {\begin{array}{*{20}{...
- 113
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
5
votes
0
answers
974
views
$\sum_{n=1}^{\infty}\frac{1}{a_n}=\infty$ $\sum_{n=1}^{\infty}\frac{1}{b_n}=\infty$ but $\sum_{n=1}^{\infty}\frac{1}{a_n+b_n}=c, c\in R$ [closed]
The following question is inspired from: Defining the slowest divergent series.
Let $a_n$ and $b_n$ be two strictly increasing sequences of natural numbers,with $\sum_{n=1}^{\infty}\frac{1}{a_n}=\...
- 3,866