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10 votes
2 answers
735 views

A number sequence problem involving binomial transform

Let $\{b_n\}_{n\geq0}$ be a sequence such that $b_nb_{n+1}=0$ and define $$a_n:=\sum_{k=0}^n(-1)^{n-k}\binom{n}{k}b_k.$$ If $\lim_{n\to\infty}a_n=0$, can we conclude that $b_n=0$ for all $n$? More ...
3 votes
1 answer
92 views

Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$

Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $a(n,m)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)=m$. ...
0 votes
0 answers
130 views

What can we say about the following number sequence?

$\{b_n\}_{n\geq0}$ is a number sequence satisfying the following condition: \begin{equation} b_{m}=\sum_{r=0}^m\sum_{h=0}^r\left(\frac{m!}{(m-r)!(r-h)!h!}\right)^2b_{m+h-r}b_{r},~\forall m\in\...
2 votes
1 answer
128 views

Is there a way to find all number series whose formulae of general term contain progressions?

Let $\{c_{m,n}\}_{m,n\in\mathbb{N}}$ be known complex numbers. My question is, how to find all number series $\{a_{n}\}_{n\in\mathbb{N}}$ such that $$a_n=\sum_{m=0}^\infty c_{m,n}a_{m+n},~\forall n\...
1 vote
1 answer
181 views

On the sequence $a(n)=\gcd(2^n-1,\phi(2^n-1))$

For natural $n$, define the sequence $$ a(n)=\gcd(2^n-1,\phi(2^n-1)) $$ It doesn't appear to be in OEIS and starts $1,1,1,1,9,1,1,1,3,1,9,1,3,1,1,1,27,1,75,49$ Q1 Can we unconditionally prove $a(n)=1$...
3 votes
1 answer
165 views

Are there infinitely many nonzero Euler quotients $a(n)=\frac{2^{\phi(n)}-1}{n} \bmod n$?

This might be related to an open problem. For odd natural $n$ define the Euler quotient: $$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$ Q1 Are there infinitely many $n$ ...
6 votes
1 answer
402 views

Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$

For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not. ...
2 votes
1 answer
153 views

Bounds for the sequence $a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$

Related to the question about a(n)=a(n-1)+a(floor(n/2)) Let $A$ be real constant $ 0 < A < 1$. Define the sequence $a(n)$ by $a(1)=1, a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$ (if you prefer take $a'...
1 vote
0 answers
194 views

Closed form for partial sums of A103318

Let $a(n)$ be A103318, number of solutions $i$ in range $[0,n-1]$ to $i \equiv 0 \pmod {2^{n-i}}$: the sequence begins with $$1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2$$ Also let's ...
2 votes
1 answer
205 views

Difference sequences of sets of integers

In this paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are ...
1 vote
0 answers
153 views

A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
3 votes
1 answer
159 views

Limit associated with two Beatty sequences that are not a Beatty pair

Suppose that $r>1$ and $s>1$ are irrational numbers, and let $a_n=\lfloor nr \rfloor$ and $b_n=\lfloor ns \rfloor$. Assume that $r$ and $s$ are numbers for which $\{a_n\}\cap\{b_n\}$ is ...
4 votes
1 answer
219 views

Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$

Note: Posting in MO since it was unanswered in MSE Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
3 votes
0 answers
195 views

Is this sequence always periodical?

Is the following sequence always periodical?
2 votes
0 answers
108 views

How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?

Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...
1 vote
1 answer
594 views

Polynomials, $3^x$ and the Collatz conjecture

$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open. Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...
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 ...
-3 votes
1 answer
544 views

Why do we need to represent integers as the sum of three cubes? [closed]

It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to $$ a^3+b^3+c^3=k. $$ Some cases for integer $k$ becomes too hard like $42$ which it ...
2 votes
1 answer
222 views

Euler quotients modulo $n$

For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$: $$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$ $a(n)=0$ for OEIS sequence Wieferich numbers ...
0 votes
1 answer
492 views

New experiments involving Ramanujan primes: Benford's law

I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the Ramanujan Prime from the online encyclopedia Wolfram MathWorld. This week I ...
5 votes
1 answer
184 views

Integer sequences with unique $k$-subsets sum

let the $\binom{\mathfrak{M}}{k}$ be a shorthand notation for chosing $k$ elements of set $\mathfrak{M}$ of positive integers and let $\left|\binom{\mathfrak{M}}{k}\right|$ denote the sum of the ...
4 votes
1 answer
217 views

Why do convoluted convolved Fibonacci numbers pop up from this triangle?

Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version): $$ T(0,0) = 2;T(1,0) = 1;...
3 votes
1 answer
308 views

Tangent numbers, secant numbers and permanent of matrices

Inspired by Question 402572, I consider the permanent of matrices $$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$ where $n$ ...
2 votes
1 answer
146 views

On gaps in a sequence of integers

Given a fixed $p \in \{3,4,5,\ldots\}$, we define the strictly increasing sequence $\{a_k\}_{k\in \mathbb N}$ as follows. We set $a_{p,1}=1$ and for each $k>1$, we set $a_{p,k}$ to be the least ...
8 votes
1 answer
363 views

Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)

I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in: 1 Bilu, Hanrot, and ...
8 votes
1 answer
364 views

Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by $$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$ In 2004, R. Chapman [Acta ...
6 votes
5 answers
546 views

Bounds for $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$

This is related to problem in graph theory. OEIS defines A033485 as $a(1)=1$ and $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$. Q1 what are upper bounds and asymptotics for $a(n)$, can we get $\exp(o(n))$? ...
1 vote
1 answer
128 views

Bounds for the sequence $a(n,A)=n*a(\lfloor (1-A)n \rfloor,A)$

Related to this question and possibly the open problem of the exponential time hypotheses. Let $A$ be rational number, $0 < A < 1$. For positive integer $n$, define the sequence $a(1,A)=1$ and $(...
6 votes
0 answers
286 views

Does $a_{i}(n)$ hit every positive integers infinitely many times for all $i\ge1$?

This question is related to a family of sequences. I have a simple definition as below and I have a question based on my limited observations for $i\le200$ and $n \le 10^{9}$. Definition. $a_{i}(1) = ...
2 votes
0 answers
176 views

A question on $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$

Let $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$ where $\pi(n)$ is the prime-counting function. By definition, it is obvious that $a_1(n) = n$ and $a_2(n)$ is https://oeis....
6 votes
0 answers
284 views

Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?

Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203) It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) ...
12 votes
1 answer
427 views

Subwords of the infinite Fibonacci word

Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS. Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be ...
3 votes
0 answers
285 views

Catalan numbers, Pochhammer symbols, Stirling numbers of the second kind, and sums of aliquot parts

For integers $N\geq 1$ we define $$s(N)=\sigma(N)-N$$ the aliquot sum function, where $\sigma(N)=\sum_{1\leq d|N}d$ is the sum of divisors function. Here $(x)_n$ is the Pochhammer symbol and ${a\...
7 votes
0 answers
945 views

Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as $$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...
5 votes
1 answer
310 views

In the Oldenburger-Kolakoski sequence, is #1s = #2s infinitely many times?

The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence: $$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$...
3 votes
1 answer
240 views

The sequence $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$

Related to this question. Let $p$ be prime and $n$ positive integer. Define $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$ Let $D(n)$ be the base $2$ discrete logarithm of $a(n)$, i.e. given $p,a(n)$ we have $2^...
5 votes
1 answer
359 views

Discrete logarithm and the sequence $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$

Let $p$ be prime and $g,n$ integers. Define $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$ By mod p we don't mean congruence, but the reduction modulo $p$ operator. $A \bmod ...
2 votes
0 answers
137 views

Writing integers as sequences of products by 2 and integer divisions by 3

For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$. For instance: $$ 100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
18 votes
2 answers
992 views

A conjecture harmonic numbers

I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven). From the Online Encyclopedia of Positive Integers we have: $a(n)$ ...
3 votes
1 answer
138 views

Properties of a certain sequence

During research I came to the following sequence: Let $\lambda>1$ and define $n_{k+1}=\text{IntergerPart}[\lambda\cdot n_k]$ where we assume that $n_0$ is sufficently large integer, so that the ...
26 votes
1 answer
3k views

A surprising conjecture about twin primes

Just for fun, I began to play with numbers of two distinct ciphers. I noticed that most of the cases if you consider the numbers $AB$ and $BA$ (written in base $10$), these have few common divisors: ...
4 votes
1 answer
435 views

Quadratic progressions with very high prime density

In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this: The list of sequences $f_k(x)$, each one corresponding to a specific $k$, ...
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}...
69 votes
1 answer
4k views

Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?

I've discovered what I believe is a quite remarkable sequence (A318970), defined by $$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$ Here are the first four terms with their prime ...
1 vote
0 answers
37 views

Raggedness measure of a sequence

This surely has been done, maybe I googled the wrong adjective... Define a raggedness measure $r$ of a sequence $S$ in this way: Two members $S_i,S_j$ of the sequence (who don't have to be adjacent!) ...
1 vote
0 answers
108 views

Question related to sequence of recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$

Define radical of an integer Wiki $$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$ Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{...
2 votes
0 answers
327 views

Why can one compute the sum of divisors of $n$ without factoring $n$?

Question links to paper which states: $$ \sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1) $$ where $\sigma(n)$ is the sum of divisors of $n$. Another similar ...
1 vote
0 answers
151 views

On smoothness and roughness of a number related to triangular numbers

Define $\triangle_n$ to be the $n$th triangular number. Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$ Define $(\ell,k)$-smough numbers to be numbers that ...
4 votes
0 answers
300 views

On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$

My question is related to https://oeis.org/A269839. It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
2 votes
2 answers
422 views

Why are attempts to define chaos with discrete states so scarce?

Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by Golomb (1991) and Tanny (1992). And in literature, there are very few studies that have different ...