All Questions
Tagged with integer-sequences nt.number-theory
241 questions
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 $\operatorname{wt}(k)=m$. ...
- 4,948
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 ...
- 65
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
$$a_n=\sum_{m=0}^\infty c_{m,n}a_{m+n},~\forall n\...
- 111
2
votes
0
answers
115
views
Closed form for the sum of the integer coefficients
Let $a(n)$ be A002720, i.e., number of partial permutations of an $n$-set; number of $n \times n$ binary matrices with at most one $1$ in each row and column.
$$a(n)=\sum\limits_{k=0}^{n} k!\binom{n}{...
- 4,948
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) > ...
- 5,470
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 ...
- 13.9k
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\...
- 111
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 ...
- 4,948
7
votes
4
answers
1k
views
Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$
Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$
(The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Volume 1 of the Lviv ...
- 4,535
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 ...
- 995
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 ...
- 479
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 ...
1
vote
0
answers
67
views
Recurrence for permutation of A007306 (denominators of Farey tree fractions)
Let $a(n)$ be A071585, i.e., numerator of the continued fraction expansion whose terms are the first-order differences of exponents in the binary representation of $4n$, with the exponents of $2$ ...
- 4,948
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
...
- 25.4k
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 $n$ ...
- 884
0
votes
0
answers
110
views
What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?
Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...
- 25.4k
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: ...
- 363
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 $m>1$ ...
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):
$$
T(0,0) = 2;T(1,0) = 1;...
- 194
32
votes
0
answers
2k
views
A question related to the Hofstadter–Conway \$10000 sequence
The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
- 701
3
votes
0
answers
195
views
13
votes
2
answers
2k
views
Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...
- 15.6k
1
vote
0
answers
70
views
Is this factorization problem in EXP?
Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored.
However now consider integers of form $...
- 13.9k
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 ...
- 479
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 ...
- 4,115
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 ...
- 13.2k
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$...
- 523
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^...
- 25.4k
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)}.$$
In 2004, R. Chapman [Acta ...
- 15.6k
5
votes
1
answer
310
views
In the Oldenburger-Kolakoski sequence, is #1s = #2s infinitely many times?
The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence:
$$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).$...
- 479
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) = ...
- 701
20
votes
2
answers
1k
views
A possibly surprising appearance of $\sqrt{2}.$
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
- 3,443
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 ...
- 25.4k
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 ...
- 34.6k
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
$k$, ...
- 3,098
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 $(...
- 25.4k
49
votes
4
answers
4k
views
Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
- 3,004
14
votes
1
answer
835
views
Special configurations on a circle from a homological algebra problem
Here is the short version of the combinatorial problem:
Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
- 26.5k
10
votes
1
answer
589
views
XOR-free sets: Maximum density?
It is known that sum-free
subsets of $\mathbb{N}$ can have
natural density at most
$\frac{1}{2}$. This density is achieved by the odd numbers: the sum of two
odd numbers is even.
I ask now a similar ...
- 151k
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) ...
- 701
5
votes
2
answers
1k
views
Proof that $3^ns + \sum_{k=0}^{n-1} 3^{n-k-1}2^{a_k}=2^m.$
How would I go about proving the following:
For any odd positive integer $s$, there exists a sequence of nonnegative integers $( a_0, a_1, \cdots, a_{n-1})$ and a nonnegative integer $m$ such that,
$...
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 ...
- 589
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....
- 701
6
votes
1
answer
240
views
On the growth and bounds for a certain sequence of integers known as Bogotá numbers
A Bogotá number is a non-negative integer equal to some smaller number, or itself, times its digital product, i.e. the product of its digits. For example, 138 is a Bogotá number because 138 = 23 x (2 ...
- 3,717
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 ...
- 701
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 ...
- 25.4k
5
votes
2
answers
393
views
What is this sequence counting?
While solving (a system of) a system of linear equations level-by-level recursively, I am finding some redundant equations for level $n\geq5$. The reason why the redundancies arise is because $P(n)\...
- 153
1
vote
1
answer
334
views
Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$?
I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^...
19
votes
1
answer
1k
views
Is every sequence that looks like an AP really an AP?
Caveat: I am not at all a number theorist, and I randomly came up with the following question while I was hiking. But I already asked two serious number theorists, and since they did not know the ...
- 11.9k
2
votes
0
answers
137
views
Writing integers as sequences of products by 2 and integer divisions by 3
For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$.
For instance:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
- 1,444