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54 votes
Accepted

A remarkable almost-identity

Consider $F(z) = 4/(3+\exp(4z))$ as a function of the complex variable $z$. It is meromorphic and has simple poles where the denominator vanishes. Namely when $4z = \log 3 + (2k +1)\pi i$ for ...
Lucia's user avatar
  • 43.5k
46 votes

Examples of integer sequences coincidences

A historical example, in the sense that the conjectural equality has been refuted: A180632 (Minimum length of a string of letters that contains every permutation of $n$ letters as sub-strings) was ...
39 votes
Accepted

Parity of the multiplicative order of 2 modulo p

This problem was asked by Sierpiński in 1958 and answered by Hasse in the 1960s. For each nonzero rational number $a$ (take $a \in \mathbf Z$ if you wish) and each prime $\ell$, let $S_{a,\ell}$ be ...
KConrad's user avatar
  • 49.9k
37 votes
Accepted

Elegant recursion for A301897

Here is an expanded version of the generating function argument I sketched in a comment. For $i=1,2,3$, define the generating functions $F_i(x,y) := \sum_{n=0}^\infty \sum_{q=0}^\infty R(n,3q+i) x^n y^...
Terry Tao's user avatar
  • 110k
32 votes
Accepted

Integrality of a sequence formed by sums

Let $A(x) = \sum_{n=1}^\infty a_n x^n$ and let $$S(x) = \sum_{k=0}^\infty (7k+8)\frac{(3k+1)!}{k!\,(2k+3)!} x^k.$$ Then the formula for $a_n$ gives $A(x) = R(x)S(x)$, where $$R(x) = \frac{1}{3}\biggl(\...
Ira Gessel's user avatar
  • 16.5k
30 votes

Examples of integer sequences coincidences

[EDITED] The classic example is A000396: "Perfect numbers n: n is equal to the sum of the proper divisors of n" and A000668(n)*(A000668(n)+1)/2 where A000668 are the Mersenne primes. They are the ...
29 votes
Accepted

A surprising conjecture about twin primes

Suppose $n-1$ and $n+1$ are both primes. $\gcd(an+b,bn+a)$ divides $an+b - (bn+a) = (a-b)(n-1)$. There are two cases. If $n-1$ divides $\gcd(an+b,bn+a)$ then $b=n-1-a$ so $an+b= (n-1) (a+1)$ and $bn+...
Will Sawin's user avatar
  • 138k
23 votes
Accepted

A finite alternating sum

I have obtained a formula for the generating function of your sequence. Let $S_n$ be defined as in the quesion. We extend the definition to $n = 0$ by demanding $0^0 = 0$, hence $S_0 = 0$. Consider $...
WhatsUp's user avatar
  • 3,232
23 votes
Accepted

Does this sequence ever end?

This is John H. Conway's Climb to a prime problem. Conway originally conjectured that no matter what number you start with, you will eventually converge to a prime. This conjecture was disproven by ...
Timothy Chow's user avatar
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22 votes
Accepted

Is every sequence that looks like an AP really an AP?

For each $n$, the differences $a_{n+1}-a_n$, $a_{n+2}-a_{n+1}$, and $a_{n+2}-a_n$ can only be divisible by powers of $2$ and primes less than or equal to $c$. Since $$ \frac{a_{n+2}-a_{n+1}}{a_{n+2}-...
Julian Rosen's user avatar
  • 8,961
20 votes
Accepted

Kindda-Perfect number: Is there a sequence of numbers which are equal to the sum of its proper divisors excluding itself as well as 1?

Such an abundant number with abundance 1 is called a quasiperfect number (which is a more professional way to say "kindda-perfect"). None have been found, according to Wikipedia. This 1982 article ...
Carlo Beenakker's user avatar
19 votes

A possibly surprising appearance of $\sqrt{2}.$

Let's define two auxiliary sequences $c_n=a_{n+2}-a_{n+1}-2$ and $d_n=b_{n+2}-b_{n+1}$ for $n\geq 1$. One can prove with an induction argument that the sequence $c_n$ takes values in $\{0,1,2,3\}$ and ...
Gjergji Zaimi's user avatar
19 votes
Accepted

Reference request: a tale of two mathematicians

Here is a recent talk by Ringel (in German): Algebra und Kombinatorik. The related part is: Besonderes Aufsehen hat ein Ergebnis erregt, das meist Satz von Gabriel genannt wird: Ein zusammenhängender ...
Mare's user avatar
  • 26.2k
18 votes

A finite alternating sum

WhatsUp's generating function allows to get an asymptotics: we have $$\sum S_n t^n=\frac{t}{e^{t/e}-t},\sum S_n e^nt^n=\frac{t}{e^{t-1}-t}.$$ Denote $t-1=x$, then $$\frac{t}{e^{t-1}-t}=\frac{1+x}{e^x-...
Fedor Petrov's user avatar
16 votes
Accepted

are these polynomials or rationals functions?

This is response to QUESTION 1. As Fedor pointed out, we're dealing with the Chebyshev polynomials $P_n(2\cos t)=\sin nt/\sin t$. So we must show that if $$ \sum_{n=1}^N \sin nt = 0 , \quad\quad\quad\...
Christian Remling's user avatar
16 votes

Series and sequences in physical systems & closed form expressions

The Casimir effect is a manifestation of $$1+2^3+3^3+\cdots=-\frac{1}{120}.$$ The vacuum energy $E$ in the space between two metal plates, separated by a distance $a$ equals $$E = \frac{ \hbar c \pi^...
15 votes

Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

I'm not quite sure what you mean by an "analytic formula." As Fedor Petrov indicated, there is unlikely to be a closed formula. However, there is a convergent power series. More precisely, consider ...
Joe Silverman's user avatar
15 votes

A conjecture harmonic numbers

We can use the characterization by Christie. Let $\pi \in S_n$. Add a fixed point $0$ to $\pi$, and let $c$ be the cycle $(0, 1, \ldots, n)$. Then the smallest number of block interchanges to sort $\...
Mikhail Tikhomirov's user avatar
15 votes

Distance among integer set

To simplify notation define $\delta(x,y) = |x-y|$. Here's an easier proof (using only Euclid) of the result in the answer that Seva posted: a finite set of positive integers closed under $\delta$ a ...
Noam D. Elkies's user avatar
14 votes

Examples of integer sequences coincidences

Just another instance of the (second) Strong Law of Small Numbers: We have A157656(n) = A059100(n-1) for all known terms (i.e., $n\leq 6$), but it's also known that A157656(29) > A059100(28). So, the ...
14 votes

Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

This is a second answer with a somewhat different viewpoint from my other answer. It is an expansion, in some sense, of Gerry Myerson's answer. There is a general theory for estimating these sorts of ...
Joe Silverman's user avatar
14 votes
Accepted

When is $\mathrm{gcd}(k,p^k-1)=1$ true?

It is easier to describe non-good (bad) numbers with respect to a given prime $p$. For each such number $k$, there exists a prime $q$ such that $q\mid k$ and $q\mid (p^k - 1)$. It follows that $k$ is ...
Max Alekseyev's user avatar
13 votes

Computationally challenging integer sequences

I think the question is too broad. There are a variety of such examples, and it seems that many combinatorial items and computational complexity items would fit the bill. The OEIS has a "most ...
Gerhard Paseman's user avatar
13 votes

Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$

2-4-6-8, this we don't appreciate! There is a long tradition of exploring additive representations of integers by polynomial sequences. The gold standard for measuring such progress is Waring's ...
Lucia's user avatar
  • 43.5k
13 votes
Accepted

XOR-free sets: Maximum density?

Let $a\in S$ be some fixed element. Note that $a\oplus b \le a + b$. Let $N$ be some big number. Put $M = [1, \ldots , N]\cap S$. We have $a\oplus M \cap M = \varnothing$. We also have $a\oplus M \...
Aleksei Kulikov's user avatar
12 votes
Accepted

Arbitrarily many primes in a Fibonacci-type sequence

I think the answer to this question is yes. The theorem of Green, Tao and Ziegler says that a collection of $K$ linear forms over the integers will all take prime values infinitely often provided that ...
Ben Green's user avatar
  • 4,766
12 votes

Limit associated with complementary sequences

Let $\alpha_*$, $\alpha^*$ denote the lower, respectively upper asymptotic density of the set $A$, and $\beta_*$, $\beta^*$ the lower and upper asymptotic density of the set $B$. Note that $$\...
Pietro Majer's user avatar
12 votes

What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

Let us denote the left hand side of $(1)$ by $\psi(x)$. It is known that $|\psi(x)-x|$ is not bounded by a constant times $x^{1/2}$. In fact Littlewood (1914) proved that $$\psi(x)-x=\Omega_{\pm}(x^{1/...
GH from MO's user avatar
  • 101k
12 votes
Accepted

$\pi(x+200)-\pi(x)\leq 50$?

Yes. Up to $207$ there are $46$ primes. Hence, the inequality is true for $x \le 7$. Let $$\pi_{210}(x) = \textrm{card}(\{n \in [0,x] \cap \mathbb{N}, \, \gcd(n,210)=1\}).$$ For $x>7$, $\pi(x+200)-\...
Ofir Gorodetsky's user avatar
12 votes

Six consecutive positive integers with certain shape

I'd like to add something from the viewpoint of heuristics. The most restrictive condition in this problem seems to be "occurence of a number of $p$-adic valuation $1$ for a small prime $p$" ...
Joachim König's user avatar

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