# Tag Info

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 ...
• 43.5k

### 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 ...
Accepted

### Parity of the multiplicative order of 2 modulo p

This problem was asked by SierpiĆski 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 ...
• 49.9k
Accepted

Here is an expanded version of the generating function argument I sketched in a comment. For $i=1,2,3$, define the generating functions $F_i(x,y) := \sum_{n=0}^\infty \sum_{q=0}^\infty R(n,3q+i) x^n y^... • 110k 32 votes Accepted ### Integrality of a sequence formed by sums Let$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 $$R(x) = \frac{1}{3}\biggl(\... • 16.5k 30 votes ### Examples of integer sequences coincidences [EDITED] The classic example is A000396: "Perfect numbers n: n is equal to the sum of the proper divisors of n" and A000668(n)*(A000668(n)+1)/2 where A000668 are the Mersenne primes. They are the ... 29 votes Accepted ### A surprising conjecture about twin primes Suppose n-1 and n+1 are both primes. \gcd(an+b,bn+a) divides an+b - (bn+a) = (a-b)(n-1). There are two cases. If n-1 divides \gcd(an+b,bn+a) then b=n-1-a so an+b= (n-1) (a+1) and bn+... • 138k 23 votes Accepted ### A finite alternating sum I have obtained a formula for the generating function of your sequence. Let S_n be defined as in the quesion. We extend the definition to n = 0 by demanding 0^0 = 0, hence S_0 = 0. Consider ... • 3,232 23 votes Accepted ### Does this sequence ever end? This is John H. Conway's Climb to a prime problem. Conway originally conjectured that no matter what number you start with, you will eventually converge to a prime. This conjecture was disproven by ... • 79.4k 22 votes Accepted ### Is every sequence that looks like an AP really an AP? For each n, the differences a_{n+1}-a_n, a_{n+2}-a_{n+1}, and a_{n+2}-a_n can only be divisible by powers of 2 and primes less than or equal to c. Since$$ \frac{a_{n+2}-a_{n+1}}{a_{n+2}-... • 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 ... • 180k 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 ... • 85.2k 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 zusammenhĂ€ngender ... • 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 $$\frac{t}{e^{t-1}-t}=\frac{1+x}{e^x-... • 104k 16 votes Accepted ### are these polynomials or rationals functions? This is response to QUESTION 1. As Fedor pointed out, we're dealing with the Chebyshev polynomials P_n(2\cos t)=\sin nt/\sin t. So we must show that if$$ \sum_{n=1}^N \sin nt = 0 , \quad\quad\quad\... • 23.4k 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 ... • 46.3k 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 \... • 5,410 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 ... • 77.8k 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 ... • 46.3k 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 ... • 31.3k 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 ... 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 ... • 43.5k 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 \... • 5,471 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 ... • 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$$\... • 57k 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 $$\psi(x)-x=\Omega_{\pm}(x^{1/... • 101k 12 votes Accepted ### \pi(x+200)-\pi(x)\leq 50? Yes. Up to 207 there are 46 primes. Hence, the inequality is true for x \le 7. Let$$\pi_{210}(x) = \textrm{card}(\{n \in [0,x] \cap \mathbb{N}, \, \gcd(n,210)=1\}).$$For$x>7$,$\pi(x+200)-\...
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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 $p$" ...