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3
votes
1answer
117 views

Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why ...
1
vote
2answers
67 views

Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...
-4
votes
0answers
150 views

What area of maths have I reinvented? [on hold]

I was trying to solve the issue of calculating (total) utilities with infinite numbers of people. This is problematic, because many infinite series are divergent, so there does not appear to be a way ...
-3
votes
0answers
49 views

Does this numerical series have any special name? [on hold]

I don't have enough background to find the answer for this on my own, so I am posting it here with the hope to get some pointers. Assume a descending sequence of K numbers $\{n_1, ..., n_K\}$ where ...
3
votes
1answer
230 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
9
votes
1answer
423 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: ...
5
votes
1answer
441 views

Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense. OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3] Numerical evidence suggest that up to $10^5$ $$ ...
9
votes
0answers
572 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 ...
3
votes
2answers
386 views

Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181: 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ... Primes $p$ ...
2
votes
1answer
439 views

Expression for infinite sum of two Bessel functions and a power

I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers): $$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} ...
1
vote
3answers
178 views

Hypergeometric sum specific value

How to show? $${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$ It numerically is very close, came up when evaluating: $$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...
3
votes
1answer
474 views

What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question) I think I understood the concept of fractional derivatives applied to ...
0
votes
1answer
113 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
4
votes
1answer
559 views

How to find the coefficients of a poor-converging series?

I have the series $\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$ and the boundary conditions $\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...
4
votes
1answer
612 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 ...
0
votes
1answer
73 views

Estimate $\left|\sum_{n,m}a_n \bar b_m\right|\leq C \left(\sum_n|a_n|^2\right)^{1/2} \left(\sum_n|b_n|^2\right)^{1/2}$ [closed]

It is well-known the Hilbert's inequality for double sum: $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. I am ...
0
votes
0answers
21 views

bounds on $\sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)}$

During our search of real rooted entire function approximations to Riemann $\Xi$ function, we need to calculate the upper and lower bounds of $$f_m(x):=\sum_{j=1}^m ...
21
votes
2answers
5k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
2
votes
0answers
73 views

Generalized Equal Distribution Kolakoski Sequence Conjecture

If we let $\operatorname{Kol}(a_1,\dots,a_n)$ be the run sequence determined by the rules of Kolakoski Frequencies, we ask is there a sequence of $\operatorname{Kol}$ that DOES NOT obey the $1/n$ ...
6
votes
1answer
168 views

Simplifying Root of Unity Double Summation

Good afternoon. I have a particular summation, $$\zeta_{n,k}(N)=\frac{k!}{N^{n+1-k}}\sum_{j=0}^n\sum_{i=0}^{N-1}\binom{n}{j}w_N^{(j-k)i}$$ Here, the $w_N$ is the root of unity ...
3
votes
1answer
207 views

How do I evaluate this sum for $s$ is a complex variable :$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$?

This question related to this question in SE ,I would like to know how do I evaluate this sum for $s$ is a complex variable :$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$$ . Edit01:And I think ...
7
votes
3answers
261 views

Upper bound on length of addition chain

An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...
6
votes
3answers
514 views

Is there a closed formula for the generating function of some trinomial coefficients?

We learn in calculus how to obtain a sum of binomial coefficients $\frac{(2d)!}{(d!)^2}$ in terms of a generating function $\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$ by the Taylor series of ...
0
votes
0answers
120 views

Difficulty understanding equivalent statement of Erdős Discrepancy Problem

Recently I watched a famous youtube video of talk given by Terry Tao on Erdős Discrepancy Problem https://www.youtube.com/watch?v=QauoO0j9Y9Y. I never heard of this problem before his announcement of ...
8
votes
1answer
280 views

Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
2
votes
0answers
56 views

Proving convergence is impossible for a sum of hyperbolic cosines

Suppose that $z$ is some complex value. Is it possible to prove that $$\lim_{n \rightarrow \infty} \sum_{j = 1}^n {\sqrt{n \over j}} \cdot \cosh(z \log {n \over j}-\operatorname{ Arccoth} (2z)) $$ ...
4
votes
3answers
523 views

Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it ...
33
votes
3answers
1k views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit ...
0
votes
1answer
107 views

An increasing sequence of real numbers [closed]

This was first posted to SE, but now I think its better to be posted here. For what positive real numbers $\alpha$, the sequence $a_n = \frac{\lfloor n\alpha\rfloor}n $ is (not necessary strictly) ...
27
votes
3answers
795 views

A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set $$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$ What can be said about the set $M$ ...
1
vote
0answers
81 views

Is there a “complete” Sidon sequence?

A sequence of natural numbers $(a_n)$ with the property that all pairwise sums of elements are distinct is called a Sidon sequence and it is proved there are at most $s(n)\sim\sqrt n$ elements of ...
15
votes
7answers
2k views

Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
17
votes
1answer
422 views

For a linear recurrence sequence $(u_n)_{n\geq 0}$, can $\{i \mid u_i > 0\}$ be the set of Fibonacci numbers?

Question: Is there a linear recurrence sequence $(u_n)_{n\geq0}$ (on the rationals, but I would also be interested by reals) for which $\text{Pos}(u) = \{i \mid u_i > 0\}$ is precisely the set of ...
1
vote
0answers
67 views

Recurrence sequence

Is it possible to find a Recurrence sequence that Satisfying the following inequality $ d_{n+k}\geq \alpha ^k d_n +‎\beta‎‎^k ‎\delta‎(A,B),$‎‎ ‎‎where $0<‎\alpha‎<1, \alpha ^k+‎\beta‎‎^k\geq ...
4
votes
2answers
313 views

Why does iterated indexing avoid cycles of length 5?

Start with a permutation $s_0$ of the numbers $(1,\ldots,n)$, e.g., for $n=10$, $s_0=(8,2,1,6,9,7,10,5,4,3)$. Form $s_1$ by using the numbers in $s_0$ as indices into $s_0$. So $s_1$ is composed of ...
12
votes
2answers
833 views

On Hamkins' answer to a problem by Michael Hardy

Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum ...
15
votes
5answers
527 views

Asymptotics of a recurrence relation

The sequence $(a_n)_{n \ge 0}$ satisfies, $a_0 = a_1 = 1$ and the recursion relation: $$a_n = \sum\limits_{k=0}^{[n/2]} \frac{a_k}{(n-2k)!}$$ where, $[x]$ is the nearest integer to $x$ not exceeding ...
1
vote
0answers
45 views

Infinite product of sine function [closed]

Does this the sequence go to zero? $\Pi_{n=1}^{N}\text{sin}(2\pi n\omega)$ as N $\rightarrow \infty$ for any $\omega \in (0,1)?$ I can see this sequence is always decreasing for general $\omega$. ...
3
votes
2answers
150 views

The minimal growth rate of the countable family of sequences

Let us consider a countable set of sequences of positive numbers $\{(x_n^{(1)}),(x_n^{(2)}),\dots\}$ for which we have $(\forall k\in\mathbb{N}) \ \lim_n x_n^{(k)} = +\infty$, and $(\forall ...
4
votes
1answer
202 views

Summation of an infinite q-series

When calculating a Partition function, I encounter the following summation $$\sum_{n=0}^{\infty} x^n q^{n^2}.$$ I know that the sum $\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I ...
3
votes
1answer
90 views

Number and asymptotic for cyclic sequences

Cyclic sequence is equivalence class of cyclic shift action. If $a = (a_1, ... , a_i)_c$ is cyclic sequence then $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, ...
1
vote
0answers
26 views

Importance sampling for bernoulli-sequence, favouring long sequences of ones

Assume we have a sequence of i.i.d. bernoulli-distributed random variables of length $n$. I'm interested in doing rare event simulation and my event depends, among other random factors, on the ...
1
vote
2answers
217 views

When is there a closed form for $\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$?

This is a follow up on a previous question of mine. Out of curiosity, I am wondering more generally when a closed form exists for $$\sum_{n=1}^{\infty} \frac{P(n)}{Q(n)}$$ where $P$ and $Q$ are ...
0
votes
0answers
48 views

Multivariate recurrence relation

Consider the recurrence relation \begin{equation} \mathbb{I}(m<M)[- k_{\rm on} c_{m,n} + (m+1) k_{\rm off} c_{m+1,n}] + \mathbb{I}(m>0)[-k_{\rm off} m \, c_{m,n} + k_{\rm on} c_{m-1,n} ] + ...
9
votes
2answers
389 views

Closed form for $\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$

I am looking at a certain sequence, and consequently I am wondering if anyone knows if there happens to exist a closed form solution for this sum: $$\sum_{n=1}^{\infty} \frac{1}{1+n+n^2+\cdots+n^a}$$ ...
8
votes
1answer
361 views

Do the complex zeros of the sum/difference of these series all reside on the line $\Re(s)=\frac12$?

The following series seems convergent for all $s\in \mathbb{C}$: $$\displaystyle f(s):=\sum_{n=1}^\infty \frac{(-1)^n}{(n+s)^{n+s}}$$ The function itself does not appear to have any real or complex ...
4
votes
4answers
461 views

What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
12
votes
1answer
280 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 ...
74
votes
8answers
6k views

Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$? (I think it is.) If so, how would one prove this? (To confirm: This is the power series for $e^x$, except with the ...
2
votes
1answer
91 views

Asymptotic upper bound for recursive function $f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$

I'm looking for an asymptotic upper bound for the function f(x), which is recursively defined as follows: $$f(x) = f(x-1) + 2f\left(\lceil\frac{x}{2}\rceil \right)+2$$ with $f(1)=1$. I am pretty ...