All Questions
Tagged with nt.number-theory sequences-and-series
236 questions with no upvoted or accepted answers
57
votes
0
answers
3k
views
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
32
votes
0
answers
2k
views
A question related to the Hofstadter–Conway \$10000 sequence
The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
- 701
28
votes
0
answers
715
views
Does this infinite primes snake-product converge?
This re-asks a question I posed on MSE:
Q. Does this infinite product converge?
$$
\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
- 151k
24
votes
0
answers
1k
views
Is A276175 integer-only?
The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
- 655
18
votes
0
answers
755
views
Two curious series for $1/\pi$
On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have
$$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$
where $(\frac{\cdot}p)$ ...
- 15.6k
16
votes
0
answers
351
views
The convergence domain of the function $\sum \{n!x\}$
This is a problem from a mathematics competition: Does there exist an irrational number $x$ such that the series
$$\sum_{n=1}^{\infty}\{n!x\}<+\infty$$
where $\{ \}$ means the fractional part of a ...
- 505
16
votes
0
answers
784
views
How to explain the picturesque patterns in François Brunault's matrix?
How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...
- 19.6k
10
votes
0
answers
350
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 2,836
10
votes
0
answers
509
views
New series for $\zeta(5)$ involving second-order harmonic numbers
In 1997 T. Amdeberhan and D. Zeilberger proved that
$$\sum_{k=1}^\infty\frac{(-1)^k(205k^2-160k+32)}{k^5\binom{2k}k^5}=-2\zeta(3).\tag{1}$$
In 2008 J. Guillera obtained that
$$\sum_{k=1}^\infty\frac{(...
- 15.6k
10
votes
0
answers
667
views
Is there a proof that OEIS-A002387 is $[ e^{n-\gamma} ]$?
Based on the comments on OEIS-A002387:
$a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,...
it is likely, that the sequence $a_{n}$ coincides with $[ e^{n-\gamma} ]$ ,
where $\gamma$ is the Euler-Mascheroni ...
user35234
9
votes
0
answers
258
views
On a continued fraction and vector $\nu$ of length $n$
Please note that this question has been completely reworked in order not to overload it with unnecessary and useless information.
Let $f(n)$ be an arbitrary function with integer values.
Let $a(n)$ ...
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9
votes
0
answers
225
views
On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...
9
votes
0
answers
498
views
Can an infinite sum depending on the logarithms of all positive integers be rational or algebraic?
Consider
$$ C = \sum_{n=1}^\infty \frac{(-1)^{f_1(n)} f_2(n) \log{n} + f_3(n)}{f_4(n)}$$
The sum converges. $f_1$ is either $0$ or $n-1$, $f_2,f_3,f_4$ are
polynomials with integer coefficients and $ ...
- 25.4k
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
0
answers
237
views
Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial
This is mostly a reference request.
I'm working with complex coefficients, although all I have in mind have integer coefficients.
Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
- 63.9k
8
votes
0
answers
197
views
The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1
We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...
- 661
8
votes
0
answers
145
views
Minimum length of sequence such that every integer from 1 to n can be achieved as the sum of some contiguous subsequence
This question literally came to me in a fever dream last night, and it's frustrating me to no end. I'll try to explain it as best I can, but there may not be a satisfying answer; the best outcome ...
8
votes
0
answers
1k
views
On the sum of consecutive primes and product of first and last
Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...
- 305
7
votes
0
answers
456
views
On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?
After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$.
I. Recurrences involving $\zeta(5)$
In Cohen's 2022 paper, ...
- 12.6k
7
votes
0
answers
429
views
Dynamics of a curious bijection of $\mathbb N$
The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows:
Given an ...
- 17.9k
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
0
answers
217
views
How to prove the identity $\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{\pi^4}{360}$?
For each $n=0,1,2,\ldots$, the harmonic number $H_n$ is given by
$$H_n:=\sum_{0<k\le n}\frac1k.$$
In 2016 I conjectured that
$$\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{...
- 15.6k
7
votes
0
answers
317
views
Alternative approaches to Zudilin's proof of Apéry's theorem
The article
Wadim Zudilin, Apéry's theorem. Thirty years after, Int. J. Math. Comput. Sci. 4 (2009), no. 1 pp 9–19. (arXiv:math/0202159, with the title An elementary proof of Apéry's theorem)
...
- 417
7
votes
0
answers
208
views
Has this self-similar sequence the ratio $(\sqrt2+1)^2$?
This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows:
$a_n$ is the smallest number such that $s_n:=...
- 13.4k
7
votes
0
answers
1k
views
A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2
Erdős asked1 whether the series
$$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges.
Here, $p_n$ denotes the n-th prime.
I can show that this series converges simultaneously with the series $\sum_{...
- 3,699
7
votes
0
answers
124
views
in search of intepretations and connections for $k$-central binomials
Fix a positive integer $k$. Then, the sequences
$$c(n,k)=\frac{k^n}{n!}\prod_{m=1}^{n-1}(1+km)=[x^n]\left(\frac1{1-k^2x}\right)^{1/k}$$
are referred to as "$k$-central binomial coefficients",...
- 43.2k
7
votes
0
answers
189
views
How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?
I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum _{n=1}^{\...
- 4,169
6
votes
0
answers
752
views
For all $n\in \mathbb{N}$, How to find $\min\{ m+k\}$ such that $ \binom{m}{k}=n$?
I asked this question on MSE here.
Most numbers in pascal triangle appear only once (excluding the duplicates in the same row of the Pascal's triangle) but certain numbers appear multiple times. ...
- 541
6
votes
0
answers
245
views
Searching for a proof of the pattern and identification of integer coefficients for the A329369
Please see the update given below. Everything you need to know from the old version of the question are the functions $a(n), \ell(n), s(n), t(n), r(n)$.
Let $a(n)$ be A329369 (i.e, number of ...
- 4,938
6
votes
0
answers
235
views
A curious series for $L(2,(\frac{-3}{\cdot}))$
Let
$$K:=L\left(2,\left(\frac{-3}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac k3)}{k^2}=\sum_{j=0}^\infty\left(\frac1{(3j+1)^2}-\frac1{(3j+2)^2}\right),$$ where $(\frac k3)$ is the Legendre ...
- 15.6k
6
votes
0
answers
298
views
A new series for $\sqrt3/\pi$?
Recently, I conjectured the following identity:
$$\sum_{k=0}^\infty\frac{(66k^2+37k+4)\binom{2k}k\binom{3k}k\binom{4k}{2k}}{(2k+1)729^k}=\frac{27\sqrt3}{2\pi}.\tag{1}$$
This can be easily checked ...
- 15.6k
6
votes
0
answers
197
views
Where to cut off a double sum?
I have to compute a double infinite sum to within a given accuracy $\epsilon$. Let us say the sum is of the form
$$\sum_{m\geq 1} \sum_{n\geq 1} \frac{a_{m,n}}{m^2 n^2 \max(m,n)},$$
where $|a_{m,n}|\...
- 20.2k
6
votes
0
answers
283
views
Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
- 161
6
votes
0
answers
3k
views
Value of $\zeta(3/2)$?
Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $\displaystyle \zeta(2)= \frac{\pi^2}{6}$ and $\zeta(3)$ has been shown to be ...
- 3,804
6
votes
0
answers
267
views
Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
- 1,755
6
votes
0
answers
97
views
Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator
This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
6
votes
0
answers
257
views
Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$
For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k$-th prime.
So multiplying by $(-1)^n$ and telescoping gives that for ...
6
votes
0
answers
136
views
On a certain $(-1)$-Eulerian polynomials of type $B$
Let $(q)_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q)_0:=1$. Define a $q$-exponential by
$$e_q(z)=\sum_{n\geq0}\frac{z^n}{(q)_n}.$$
There is a notion of $q$-Eulerian polynomials of type $A$, see the ...
- 43.2k
6
votes
0
answers
306
views
Irrationality of the sum of the reciprocal of perfect powers
A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that $$\sum_{p\in\...
- 10.9k
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
5
votes
0
answers
133
views
Formula and smallest solution for the A260711
Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$).
The sequence begins with
$$
8, 16, 32, 48, 64, ...
- 4,938
5
votes
0
answers
343
views
Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
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
5
votes
0
answers
361
views
Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series
There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas:
1) The ...
- 4,169
5
votes
0
answers
138
views
On the comparison of Egyptian fractions of two kinds
I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset \...
- 305
5
votes
0
answers
765
views
Computability of OEIS A034891 ...partitions of n into prime parts (1 included)
On the seqfan mailing list RGWv gave short algorithm for computing A000041 number of partitions of n the partition numbers:
f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == ...
- 25.4k
4
votes
0
answers
118
views
Something (which might be called multi-continued fraction) for the A112487
Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function
$$
A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\
A(0)=1
$$
However, the definition in the name of the sequence is
...
- 4,938
4
votes
0
answers
90
views
Closed form for subsequence of the partial sums of generalized A329369
Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$
Here ${n\brace k}$ is the Stirling number of the second kind.
...
- 4,938
4
votes
0
answers
279
views
What is the exact value of the series $\sum_{k=0}^\infty \binom{2k}k^4/256^k$?
By Stirling's formula $n!\sim\sqrt{2\pi n}(n/e)^n$, we have
$$\binom{2k}k\sim\frac{4^k}{\sqrt{k\pi}}$$
and hence
$$\frac{\binom{2k}k^4}{256^k}\sim\frac1{k^2\pi^2}.\tag{1}$$
So the series
$$\sum_{k=0}^\...
- 15.6k
4
votes
0
answers
167
views
Binary iterations, Fibonacci numbers and permutation of natural numbers
Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Also let's consider
$$\ell(n)=\left\lfloor\log_{2} n\right\rfloor$$
and
$$T(n,...
- 4,938