Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
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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 ...
0
votes
1
answer
296
views
Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function
For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...
0
votes
1
answer
122
views
Permutation of the natural numbers from operation related to binary expansion of $n$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let $T(n,k)$ be a $(k+1)$-th bit from the right side in the binary expansion of $n$. Here
$$
T(n, k) = \left\lfloor\frac{n}{2^k}\right\rfloor \...
- 4,948
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votes
1
answer
90
views
Reducing recurrence relations mod10 [closed]
I have been playing around with integer sequences as of late, and the following question occurred to me:
Suppose for $m$ fixed we have some some initial values $a_1,\cdots,a_m$ and for all $n\in\...
0
votes
1
answer
134
views
How to write a given rank matrix with some constraints?
If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.
...
- 13.9k
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votes
1
answer
250
views
What is a description of winning strategies in this tile game?
I'm hoping someone can help me figure out how to describe all winning strategies for "Player 1" in the following game:
Consider a board with $n$ tiles arranged in a row. Player 1 and Player 2 each ...
- 11
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1
answer
212
views
A problem inspired in the definition of tau numbers and a divisibility relationship related to powers of two
It is (I assume that this easy fact is well-known) obvious that an integer $n>1$ is a power of two $n=2^{\alpha}$, where $\alpha\geq 1$ is integer, if an only if $n$ satisfies the divisibility ...
0
votes
2
answers
226
views
sequence, such that sum of any combinations in the sequence does not equal another [closed]
Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.
...
- 119
0
votes
1
answer
140
views
Series reversion using something like continued fraction
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let
$$
F(x)=\sum\limits_{m\geqslant 0}f(m)x^m
$$
Define the operator $\operatorname{SR}$, which is associated with the series ...
- 4,948
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votes
1
answer
101
views
Recurrence for the number of steps required to get one ball in each box
Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...
- 4,948
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1
answer
220
views
Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}$ for an integer sequence?
The infinite series representation
\begin{equation}
\frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0
\end{equation}
for the positive ...
- 1,101
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votes
1
answer
144
views
Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube
The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$
\begin{equation} \label{one}
\int_0^1 \...
- 723
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votes
1
answer
61
views
Ordered $m$-tuples with fixed number of changes
Given $1\leq k\leq m$, $2\leq d\leq c i\ln i$ and $2\leq i\leq c'\ln(mi\ln i)$ at some $c,c'>0$ how many sequences (lower and upper bounds) are of form $$z_1,\dots,z_m$$ on the condition that
$$0\...
- 1,836
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1
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95
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The growth of a sequence related to Liouville numbers [closed]
I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends ...
- 9
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104
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Elementary description to count of perfect squares - II
What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$?
Is $g(a)=1\iff a=b^2+...
- 13.9k
0
votes
1
answer
133
views
Elementary description to count of perfect squares - I
Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...
- 13.9k
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215
views
Number of squares in a grid under certain conditions
Consider an $(n+1)\times (n+1)$ grid of lattice points in the plane.
$A(n):$ # of squares with vertices on the grid.
It's relatively well-known that $A(n)=\frac{n(n+1)^2(n+2)}{12}$. Now, $A(n) = B(...
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0
answers
28
views
Short periods modulo primes of linear recurrences with polynomial coefficients
Let $f_i(x)$ be polynomials with integer coefficients.
Define the integer linear recurrence with polynomial coefficients:
$$
a(n)=f_1(n) a(n-1)+f_2(n)a(n-2)+\cdots +f_d(n) a(n-d)
$$
and the initial ...
- 25.4k
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0
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55
views
Sequences that sum up to sums of integer coefficients
Let
$$
T(n,k,p,q,r,s) = (q(k-1)+1)T(n-1,k,p,q,r,s) + s(n+r(k-1)+p-2)T(n-1,k-1,p,q,r,s), \\
T(n,1,p,q,r,s) = 1, \\
T(n,0,p,q,r,s) = T(0,k,p,q,r,s) = 0
$$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\...
- 4,948
0
votes
0
answers
190
views
On a A057985 without recursion
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$).
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\...
- 4,948
0
votes
0
answers
63
views
Pairs of permutations such that $p(n)<2^k$ iff $n<2^k$
Let $p(n)$ be an arbitrary permutation of natural numbers such that $p(n)<2^k$ iff $n<2^k$.
Let $q(n)$ be an inverse permutation of $p(n)$.
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
...
- 4,948
0
votes
0
answers
107
views
Formula for individual term of the Proth numbers
Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$.
The sequence begins with
$$
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129
$$...
- 4,948
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votes
0
answers
62
views
Linear recurrences in coefficients of powers of quotients of polynomial rings
It is known that linear recurrences with constant coefficients
can be computed via powers in $\mathbb{Z}[x]/f(x)$.
We believe that this generalizes to quotients of multivariate polynomial
rings.
Let $...
- 25.4k
0
votes
0
answers
60
views
Existence of integer sequence under simultaneous constraints
Does there exist a function $f:\Bbb N\to\Bbb N$ such that \begin{align}a_{n+1}&=f(a_n)\\a_{f(n)+1}&=a_n\end{align} implies $\{a_n\}_{n\ge0}$ is a non-constant, positive integer sequence? ...
- 1,459
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0
answers
61
views
Stolarsky array and Stolarsky representation
Let $T(n,k)$ be A035506, i.e., Stolarsky array read by antidiagonals. Here we consider that $T(n,k)=0$ for $n<1, k<1$.
Let $a(n)$ be A200714, i.e., Stolarsky representation interpreted as binary ...
- 4,948
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0
answers
94
views
Closed form for the number of steps required to get $n$ balls in the last box
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Then we have an integer sequence given by
$$a(n)=n(n+1)-\sum\limits_{k=0}^{n}\...
- 4,948
0
votes
0
answers
110
views
What will be the set of non-Wieferich numbers if the set of non-Wieferich primes is finite?
Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.
Wieferich prime is Wieferich number with $n$ prime.
It is an open problem if there are infinitely many Wieferich primes
and ...
- 25.4k
0
votes
0
answers
130
views
What can we say about the following number sequence?
$\{b_n\}_{n\geq0}$ is a number sequence satisfying the following condition:
\begin{equation}
b_{m}=\sum_{r=0}^m\sum_{h=0}^r\left(\frac{m!}{(m-r)!(r-h)!h!}\right)^2b_{m+h-r}b_{r},~\forall m\in\...
- 111
0
votes
0
answers
60
views
Reference request: Counting integer sequences in homogeneous linear recurrences
Are there references in the literature that deal with the probability of finding an integer sequence in a linear homogeneous recurrence with constant coefficients $ \in \mathbb{Z}$? (or provides a way ...
0
votes
0
answers
54
views
A distribution of maximum of sums if add to the minimal
Consider a vector of $n$ integer variables with initial values of 0. Each step we take random $w_i\thicksim NB(q, l)$ (independent randon values with the same negative binomial distribution) and add ...
- 11
0
votes
0
answers
115
views
Roots of a family of 4-parameter polynomials
Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by
$$
f(x)=x^q-kx^{q-p}-\ell.
$$
This polynomial is related to a family of two-...
- 29
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votes
0
answers
86
views
Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
0
votes
0
answers
248
views
Divisibility Properties of Pisano Periods
Let $(F_n)$ the Fibonacci sequence and $\pi(m)$ the Pisano period of $m$ (i.e., the smallest period of $F_n \pmod{m}$). There are many proved results about $\pi(m)$. For example, it is known that $\pi(...
- 21
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votes
0
answers
72
views
Rewriting a set of integers to get rid of repetition but keeping subset sum ordering
Say, I have a set of 6 +ve integers sorted in ascending order:
$A = \{2,4,4,4,5,7\}$
Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them:
$\therefore B= ...
- 21
0
votes
1
answer
62
views
Mapping naturals to pairs of naturals and viceversa [closed]
I can't find much on the internet about this, but apparently vectors of naturals are called hyperscalars. It's not hard to bijectively map naturals to 2D hyperscalars and with that to prove that any-...
- 13
0
votes
0
answers
88
views
Infinite difference length of integer subsets
Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:
We say $A$ has infinite difference length, if
(a) For every integer $n$ there exist a ...
- 995
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votes
0
answers
81
views
Terminology for "approximately convex" sequence of integers
I have a sequence of integers meeting the following inequality:
$u_n \leq \frac{u_{n-1}+u_{n+1}}{2} + \frac{1}{2}$. In other words, the sequence is "approximately convex", and the difference comes ...
- 101
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votes
0
answers
93
views
What is the class of real sequences satisfying these conditions?
I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions:
$\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
0
votes
0
answers
315
views
Number Theory and p-Power-Partitioned Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...
user89024
0
votes
0
answers
540
views
Number Theory and d-Self-Contained Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...
user89024
0
votes
0
answers
224
views
Classic question on integer partitions (with distinct summands)
I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem:
Denote $R(N,L)$ ...
- 1
-1
votes
1
answer
72
views
Create approximations of finite integer sequence
Given a function of real numbers f(x), I can create approximations to arbitrary precision using Taylor polynomials.
Is there something equivalent in the discrete case when I have a sequence of ...
- 103
-2
votes
1
answer
168
views
Two-variable continuous function which results in an integer if and only if arguments are integer
I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...
- 3
-2
votes
1
answer
180
views
Decimal digits multiplied by powers of 2: leads to mod 8? [closed]
This is more a puzzle than a research question,
a puzzle to me. Perhaps it is straightforward for others.
Imagine Repeatedly interpreting a number
expressed with the usual base-$10$ digits
as "digits"...
- 151k
-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.
$$
Some cases for integer $k$ becomes too hard like $42$ which it ...
-4
votes
1
answer
178
views
Covering system of congruences with specific properties?
A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$, ($r\geq2$) is called a covering system of congruences if every integer belongs to at least one of the residue classes, ...
- 841
-4
votes
1
answer
250
views
What are the patterns of the sequence of polynomials? [closed]
In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them):
\begin{align}
& f(2) = 1+t, \\
& f(3) = 1+4t+3t^2, \\
& f(4) = 1+6t+12t^2+7t^3, \\
&...
- 6,211
-4
votes
2
answers
763
views
An interesting, simple, sequence - surprised to find little material. [closed]
I've been considering this sequence:
$$1,2,3,6,12,24,48,96,192,...$$
I've generated the sequence from the rule
$$V_n=\sum_{0\leq i \lt n} V_i$$
$$V_0=1; V_1=2V_0=V_0+V_0$$
What interests me most, ...
- 5
-5
votes
1
answer
454
views
Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]
I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...