All Questions
Tagged with nt.number-theory sequences-and-series
92 questions
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
13
votes
1
answer
782
views
Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,...$ (where $H(k)$ is the Hamming-weight)
In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
- 5,342
12
votes
4
answers
5k
views
How to calculate the infinite sum of this double series?
I'm calculating this double sum:
$$
\sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2}
$$
I know the answer is
$$
\frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16}
$$
which can be ...
- 121
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 113
10
votes
1
answer
694
views
Prime numbers from permutation
Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
- 4,938
10
votes
1
answer
434
views
Series for $\frac{\log m}{\pi}$ with summands involving harmonic numbers
The classical rational Ramanujan-type series for $1/\pi$ have the following four forms:
\begin{align}\sum_{k=0}^\infty(ak+b)\frac{\binom{2k}k^3}{m^k}&=\frac{c}{\pi},\label{1}\tag{1}
\\\sum_{k=0}^\...
- 15.6k
8
votes
2
answers
375
views
Growth of a linear recurrent sequence
Consider the sequence defined by $a_0=a_1=1$ and $a_n=2a_{n-1}-3a_{n-2}$ for $n\geq 2$. This is the sequence https://oeis.org/A087455.
I would like to prove that $|a_n|>100$ when $n>10$. How ...
- 183
8
votes
2
answers
429
views
Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such ...
- 183
8
votes
4
answers
344
views
Simple-looking sequences $A$ and $B$ defined by a complementary equation
Define $A=(a_n)$ and $B=(b_n)$ by $b_0=1$ and
$$a_n=b_n+b_{2n}$$
for $n \geq 0$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B$. Can someone prove ...
- 3,443
8
votes
0
answers
351
views
A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$
Recently, I found a (conjectural) new series for $\sqrt3\pi$:
$$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$
The series converges fast ...
- 15.6k
8
votes
2
answers
2k
views
5n+1 sequence starting at 7
Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by
\begin{equation}
f(n):=\begin{cases}
n/2 & \text{if $n$ is even}\\
5n+1 & \...
- 654
7
votes
0
answers
226
views
Is there a connection between the sequence of a finite number of Stieltjes constants and the integer partitions number?
Lehmer 1988 and Keiper 1992 made major progress on evaluating the series:
$$\sigma_r = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right) \quad r \in \mathbb{N}$$
where $\...
- 4,169
7
votes
1
answer
240
views
$q$-Eulerian type B enjoy symmetry
Let $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q;q)_0:=1$. Define a $q$-exponential by
$$e(z;q)=\sum_{n\geq0}\frac{z^n}{(q;q)_n}.$$
There is a notion of $q$-Eulerian polynomials, see the reference. I ...
- 43.2k
6
votes
2
answers
389
views
Conjectured Somos-like closed form of recurrences with polynomial coefficients
From Our short paper
For polynomial $F$ with integer coefficients, define the recurrence
$f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that
$f(n)$ satisfy Somos like sequence
$f(n)=\frac{G(f(n-1)...
- 25.4k
6
votes
1
answer
609
views
Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers
Let $a(n)$ be the Dirichlet inverse of the Euler totient function:
$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$
and let the matrix $T(n,k)$ be:
$$T(n,k)=a(\gcd(n,k)) \tag{2}$$
It has been ...
- 1,183
6
votes
1
answer
268
views
Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$
Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$.
The sequence begins with
$$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
- 4,938
6
votes
3
answers
1k
views
Does this 'alternating' Euler product converge for all $\Re(s) > 0$?
Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} \right)^{(...
- 4,169
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
2
answers
385
views
How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?
I have to estimate the expression $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for $\kappa$ very small $\kappa \sim 10^{-19}$ and $N$ very large $N\sim 10^{26}$ and $a$ arbitrary $a=1, \ldots, N$. I do not ...
5
votes
1
answer
742
views
mertens-function in the light of divergent summation - what summation method were best adapted
Just reading about the Mertens-function in the other thread
Mertens function I remember an earlier attempt to apply divergent summation
to the series which is constructed of the Moebius-function
at ...
- 5,342
5
votes
1
answer
819
views
Is this known alternating sum for Euler's constant?
This probably is known, but Wolfram Alpha doesn't recognize it
and couldn't find it in Mathworld (there is something close,
but using floor).
We have
$\lim_{s \to 1} (\zeta(s)-1/(s-1)) = \gamma$
...
- 25.4k
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
4
votes
1
answer
628
views
Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$
Let us define the following functions:
\begin{equation*}
\small A(x)=\prod_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
B(x)=\prod_{\substack{p\leq x\\ p\...
- 3,098
4
votes
2
answers
337
views
Is this BBP-type formula for $\ln 31$, $\ln 127$, and other Mersenne numbers also true?
In this post, a binary BBP-type formula for Fermat numbers $F_m$ was discussed as (with a small tweak),
$$\ln(2^b+1) = \frac{b}{2^{a-1}}\sum_{n=0}^\infty\frac{1}{(2^a)^n}\left(\sum_{j=1}^{a-1}\frac{...
- 12.6k
4
votes
2
answers
354
views
Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$
I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
- 791
3
votes
0
answers
237
views
Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$
The $n$-th convergent is defined as
$$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$
where $P_n(x), Q(x)$ are polynomials ...
- 3,098
3
votes
1
answer
240
views
The sequence $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$
Related to this question.
Let $p$ be prime and $n$ positive integer.
Define $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$
Let $D(n)$ be the base $2$ discrete logarithm of $a(n)$, i.e.
given $p,a(n)$ we have $2^...
- 25.4k
3
votes
1
answer
138
views
Sequences that sum up to Dowling numbers
Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...
- 4,938
3
votes
2
answers
458
views
Subsequence of the cubes
Let $p$ and $q$ be integers.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then ...
- 4,938
3
votes
0
answers
151
views
Combinatorial interpretation of inverse modulo $2$ binomial transform of A284005
My question is related to the following:
Sum with products turned into subsequences
We have an identity
$$a(n, -1) = \sum\limits_{j=0}^{2^{\operatorname{wt}(n)}-1}(-1)^{\operatorname{wt}(n)-\...
- 4,938
2
votes
2
answers
655
views
Expansion of inverse logarithmic integral in terms of lambert w
Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:
$\operatorname{li}^{-...
- 1,903
2
votes
2
answers
416
views
What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?
What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
- 153
1
vote
0
answers
116
views
On level $6$ of the McKay–Thompson series of the Monster and Apéry numbers, et al
After the McKay–Thompson series of levels $1$, $2$, $3$, $4$ of the Monster were mentioned in this MO post, level $6$ has very interesting relations as well. (Level 10 is in this post.)
I. Level-6 ...
- 12.6k
1
vote
0
answers
95
views
On level-$12$ of the McKay-Thompson series of the Monster and the Domb numbers
(This continues from level 10.) Given some moonshine functions $j_{N}$. There are nice descending and consistent relations for levels $6m$ with $m= 2,3,5,$
$$j_{12A} = \left(\sqrt{j_{12H}} + \frac{\...
- 12.6k
1
vote
1
answer
344
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k
1
vote
1
answer
152
views
Hybrid numeration system on $[0,1]^2$
Let $X_0,X_1\in [0,1]$ and $b_1,b_2>0$ be integers. We are going to create a numeration system for vectors $(X_0,X_1)$, the base being the vector $(b_1,b_2)$, as follows.
Recursively define $X_k=\{...
- 3,098
1
vote
1
answer
110
views
Number of steps required to get one ball in each box for $n=2^k$
Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...
- 4,938
1
vote
1
answer
128
views
Bounds for the sequence $a(n,A)=n*a(\lfloor (1-A)n \rfloor,A)$
Related to this question and possibly the open problem
of the exponential time hypotheses.
Let $A$ be rational number, $0 < A < 1$.
For positive integer $n$, define the sequence
$a(1,A)=1$ and $(...
- 25.4k
0
votes
0
answers
109
views
Simplification of summation and reverse search
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $b(n)$ be an integer sequence such that
$$b(2^m(2n+1))=\sum\limits_{k=0}^{m}f(m-k)b(2^kn), b(0)=1$$
Let $s(n,m)$ be an integer ...
- 4,938
0
votes
0
answers
182
views
Expansion of continued fraction using recursion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where
$$
G(j)=1-\frac{f(j)x}{G(j+1)}
$$
Here we have
$$
G(...
- 4,938
-1
votes
14
answers
2k
views
Priming for the primes [closed]
I have to confess that most often my eyes begin to glaze over when someone starts discussing the prime numbers. However, my ears have perked up at times over the primes--maybe first when I learned of ...
-2
votes
1
answer
217
views
Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
- 317