All Questions
Tagged with integer-sequences nt.number-theory
241 questions
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
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
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
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
3
votes
1
answer
247
views
A special type of Langford pairing
A Langford pairing is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of ...
- 3,717
3
votes
1
answer
298
views
Sequences with integral variances
This is a companion to my earlier question,
Sequences with integral means.
This new question is, frankly, not as interesting, but it feels necessary to complete
the thought.
Let $V(n)$ be the ...
- 151k
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
3
votes
1
answer
140
views
Sequences that sum up to Dowling numbers
Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...
- 4,948
3
votes
1
answer
344
views
Another integral that has a closed form involving finite series of $\zeta(2k+1)$'s. Could it be reflexive?
In the context of a series of questions here, here and here, about closed form expressions involving finite series of $\zeta(2k+1)$'s for certain integrals, I would like to raise another one:
$$f(n):=...
- 4,169
3
votes
0
answers
128
views
Fast and simple algorithm for the A329369
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\cdots,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
- 4,948
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,948
3
votes
0
answers
69
views
Sequence that sum up to A343685
Let $a(n)$ be A343685 i.e.
$$
a(n)=2na(n-1)+\sum\limits_{j=0}^{n-1}\binom{n}{j}(n-j-1)!a(j), \\
a(0)=1
$$
Here the exponential generating function $A(x)$ satisfy
$$
A(x)=\frac{1}{1-2x+\log(1-x)}
$$
...
- 4,948
3
votes
0
answers
165
views
Closed form for $a(2^m(2^n-2^p-1))$
Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
- 4,948
3
votes
0
answers
195
views
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\...
3
votes
0
answers
180
views
Additive combinatorics and a Diophantine equation
Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
$...
- 2,227
3
votes
0
answers
86
views
Increasing integral sequence of intermediate growth which is periodic modulo almost all primes
Many integral sequences are periodic modulo (almost) all primes.
However all examples I know are either evaluations of suitable polynomials on consecutive integers (trivial examples) or grow at least ...
- 17.9k
3
votes
0
answers
100
views
Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers
Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$
$$
f(n) =
\left\{
\begin{array}{ll}
\mu\left(\frac{n}{2}\right) + \mu\left(\frac{n}{4}\right), & n\equiv 0, 4, 8\mod 12, \\
...
- 410
3
votes
0
answers
252
views
What are the values of this sequence?
Let $F_n$ denote the $n$th Fibonacci number.
Then $\prod\limits_{i=1}^{\infty}(1-x^{F_i})$ is a series all of whose coefficients are either $-1$, $0$ or $+1$.
The sequence of the coefficients in ...
- 2,143
2
votes
2
answers
197
views
On the primality of $j(n)=\varphi(p_n+1-n)+1$ when $j(n) \equiv 19 \pmod {100}$
Related to Power of primes.
Let $p_n$ denote n-th prime and $\varphi$ the totient function.
For natural $n$, define $j(n)=\varphi(p_n+1-n)+1$.
For $n$ up to $10^9$ if $j(n) \equiv 19 \pmod {100}$
then ...
- 25.4k
2
votes
1
answer
214
views
Tower-of-squares sequence divides linear recurrent A001921 sequence?
Let $(a_n)$ be the A001921 sequence
$$
a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6.
$$
Let $(b_k)$ be the (almost)"tower-of-squares" sequence defined by
$$
b_0=2, \quad b_{k+1}=2b_k^...
- 621
2
votes
2
answers
242
views
Negated Fibonacci and the floor function
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1, \\
F_{-n} = (-1)^{n-1}F_n
$$
I conjecture that
$$
F_{-n} = \left\lfloor\frac{n+1}{2}\right\rfloor ...
- 4,948
2
votes
1
answer
268
views
A problem similar to the $3x+1$-problem [closed]
Let $n$ be a fixed positive integer. Define the function $f_n(x)$ as follows:
$$f_n(x)=\left\{\begin{aligned}&2x-1,\quad x\leq n;\\&2(x-n),\quad x> n.\end{aligned}\right.$$
and for $l\in\...
- 111
2
votes
1
answer
220
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
2
votes
1
answer
740
views
Power tower made of $2$s and $3$s: too high, too soon?
A power tower of a number $x$ is typified by
$$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$
Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers
$$...
- 3,443
2
votes
1
answer
216
views
Simplification of the closed form for the A329369
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let ${n \brace k}$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace ...
- 4,948
2
votes
1
answer
261
views
Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
- 25.4k
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'...
- 25.4k
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
2
votes
1
answer
913
views
Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way [closed]
I need to emulate this sequence for a program: http://oeis.org/A025302
Stuff that I've taken into account:
After finding the prime divisors of a number. I take any divisor as p and apply the ...
- 123
2
votes
1
answer
252
views
Is there a linear recurrence with infinitely many zeros, conjecturally infinitely many primes and non-zero terms of exponential growth?
Let $a_n$ be a linear recurrence with integer constant coefficients
and initial values.
Is it possible $a_n$ to satisfy all of these:
$a_n = 0$ infinitely often.
if $a_n \ne 0$, $ | a_n |$ is of ...
- 25.4k
2
votes
1
answer
131
views
Sequence that sums up to A224071
Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here
$$
a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
- 4,948
2
votes
1
answer
532
views
The Euler's totient function and the product of distinct primes dividing $n$ versus the Heronian means
For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this Wikipedia). And $\...
user75829
2
votes
1
answer
154
views
GCD for two Cullen numbers
The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
- 21
2
votes
1
answer
120
views
Recursion for the Chebyshev transform of $m^n$
Let
$$
R(n, q, m) = R(n-1, q+1, m) + \sum\limits_{j=0}^{q} (-1)^{q-j}R(n-1, j, m), \\
R(0, q, m) = (m-1)^q
$$
I conjecture that $R(n, 0, m)$ is a Chebyshev transform of $m^n$.
Examples of Chebyshev ...
- 4,948
2
votes
1
answer
205
views
Difference sequences of sets of integers
In this paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are ...
- 995
2
votes
1
answer
238
views
"flavored" equivalence classes of permutations
We say two permutations $\pi_1$ and $\pi_2$ in the symmetric group $\mathfrak{S}_n$ are $k$-equivalent, denoted
$\pi_1 \sim_k \pi_2$, if one can be
determined from the other after a finite number of ...
- 43.2k
2
votes
1
answer
380
views
Repdigit numbers, which are sum of consecutive squares
Following up on this question,
https://math.stackexchange.com/questions/1788015/is-112122132142152162-1111-special/1788102?noredirect=1#comment3649733_1788102
is anything known about the sequence of ...
- 277
2
votes
1
answer
172
views
Permutation and its binary analog
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 4,948
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
$$a_n=\sum_{m=0}^\infty c_{m,n}a_{m+n},~\forall n\...
- 111
2
votes
2
answers
422
views
Why are attempts to define chaos with discrete states so scarce?
Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by Golomb (1991) and Tanny (1992).
And in literature, there are very few studies that have different ...
- 701
2
votes
0
answers
76
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
2
votes
0
answers
62
views
Algorithm for main diagonal of integer coefficients associated with Schroeder numbers
Let $T_q(n, k)$ be an integer table such that
$$T_q(n, k) = \begin{cases}
1 & \textrm{if } n = 0 \vee k = 0 \\
qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\
T_q(n, k-1) + T_q(n-1,...
- 4,948
2
votes
0
answers
28
views
On doubling or addition formulas for the sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
We are interested which integer sequences are efficiently computable
possibly over finite rings.
Define the integer sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
with initial terms $a(0),a(1)$...
- 25.4k
2
votes
0
answers
157
views
Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
- 25.4k
2
votes
0
answers
110
views
bijection from vectors with non-negative integer integer entries to integers
I have the following question. Given a natural number $N$ we construct a set $K$ of vectors of infinite length with non-negative integer entries with a given sum $N$. For example, for $N=3$ the set $K$...
- 61
2
votes
0
answers
163
views
Interesting conjecture by Sequence Machine
Let $a(n)$ be A344960 (i.e., position of binary complement of $n$-th word in A341258). By definition, in order to calculate $a(n)$, we need to know A341258. Below we will correspond this sequence with ...
- 4,948
2
votes
0
answers
72
views
Possible subsequence of the A110978
Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
- 4,948
2
votes
0
answers
199
views
Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 4,948
2
votes
0
answers
126
views
Recurrence for A004208
Let $a(n)$ be A004208. Here
$$a(n)=n\prod\limits_{j=1}^{n}(2j-1)-\sum\limits_{i=1}^{n-1}a(i)\prod\limits_{j=1}^{n-i}(2j-1)$$
I conjecture that
$$a(n)=R(n-1,0)$$
where
$$R(n,q)=2(q+2)R(n-1,q+1)+\sum\...
- 4,948