55
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 ...
- 43.7k
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 ...
- 50.6k
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^...
- 114k
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
$$R(x) = \frac{1}{3}\biggl(\...
- 17k
30
votes
Examples of integer sequences coincidences
[EDITED]
The classic example is A000396: "Perfect numbers n: n is equal to the sum of the proper divisors of n"
and A000668(n)*(A000668(n)+1)/2 where A000668 are the Mersenne primes.
They are the ...
29
votes
Accepted
A surprising conjecture about twin primes
Suppose $n-1$ and $n+1$ are both primes.
$\gcd(an+b,bn+a)$ divides $an+b - (bn+a) = (a-b)(n-1)$.
There are two cases. If $n-1$ divides $\gcd(an+b,bn+a)$ then $b=n-1-a$ so $an+b= (n-1) (a+1)$ and $bn+...
- 148k
23
votes
Accepted
A finite alternating sum
I have obtained a formula for the generating function of your sequence.
Let $S_n$ be defined as in the quesion. We extend the definition to $n = 0$ by demanding $0^0 = 0$, hence $S_0 = 0$.
Consider $...
- 3,432
23
votes
Accepted
Does this sequence ever end?
This is John H. Conway's Climb to a prime problem. Conway originally conjectured that no matter what number you start with, you will eventually converge to a prime. This conjecture was disproven by ...
- 82.6k
22
votes
Accepted
Is every sequence that looks like an AP really an AP?
For each $n$, the differences $a_{n+1}-a_n$, $a_{n+2}-a_{n+1}$, and $a_{n+2}-a_n$ can only be divisible by powers of $2$ and primes less than or equal to $c$. Since
$$
\frac{a_{n+2}-a_{n+1}}{a_{n+2}-...
- 9,061
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 ...
- 188k
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 ...
- 85.5k
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 ...
- 26.5k
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 $$\frac{t}{e^{t-1}-t}=\frac{1+x}{e^x-...
- 109k
16
votes
Accepted
are these polynomials or rationals functions?
This is response to QUESTION 1.
As Fedor pointed out, we're dealing with the Chebyshev polynomials $P_n(2\cos t)=\sin nt/\sin t$. So we must show that if
$$
\sum_{n=1}^N \sin nt = 0 , \quad\quad\quad\...
- 24.2k
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 ...
- 47.4k
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 $\...
- 5,590
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 ...
- 79.9k
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 ...
- 47.4k
14
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 \...
- 6,476
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 ...
- 34.3k
14
votes
Accepted
Do there exist prime numbers of the form $n \cdot 2^n + 1$, when $n \in \mathbb{N}$ and $n > 1$?
There's $n=141$, $n=4713$, etc. See Guy, Unsolved Problems in Number Theory, B20. They're called Cullen primes. It's not known whether there are infinitely many. A005849.
https://en.wikipedia.org/wiki/...
- 16.2k
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 ...
- 13k
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 ...
- 43.7k
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 $$\...
- 60.5k
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
$$\psi(x)-x=\Omega_{\pm}(x^{1/...
- 105k
12
votes
Accepted
$\pi(x+200)-\pi(x)\leq 50$?
Yes.
Up to $207$ there are $46$ primes. Hence, the inequality is true for $x \le 7$.
Let $$\pi_{210}(x) = \textrm{card}(\{n \in [0,x] \cap \mathbb{N}, \, \gcd(n,210)=1\}).$$
For $x>7$, $\pi(x+200)-\...
- 14.6k
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 $p$" ...
- 2,959
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