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Questions tagged [integer-sequences]

For questions about sequences of integers. References are often made to the online resource oeis.org.

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16 votes
2 answers
2k views

Does this sequence ever end?

This may help: A080670 A195265 Define $f(n)$ as this: Take a number $n$, and split it into its prime composition using $^$ and $×$. Now remove all $^$ and $×$, you get a new number, this is $f(n)$ (...
look at me's user avatar
6 votes
0 answers
224 views

Searching for a proof of the pattern and identification of integer coefficients for the A329369

Please see the update given below. Everything you need to know from the old version of the question are the functions $a(n), \ell(n), s(n), t(n), r(n)$. Let $a(n)$ be A329369 (i.e, number of ...
Notamathematician's user avatar
5 votes
0 answers
269 views
+50

On $s$-additive sequences

For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
Sayan Dutta's user avatar
1 vote
1 answer
150 views

Correctness of the algorithm for the A329369, A347205 and related sequences

Let $a(n)$ be A347205. It is enough for us to know that $$ a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^jk), \\ a(0) = 1 $$ Let $b(n)$ be A329369. It is enough for us to know that $$ b(2^m(2k+1)) = \sum\...
Notamathematician's user avatar
1 vote
0 answers
132 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 ...
joro's user avatar
  • 24.3k
6 votes
1 answer
343 views

Test for pair of odd primes $(p, 2p^2-1)$

Let $a(n)$ be A106483 (i.e., primes $p$ such that $2p^2-1$ is also prime). Let $b(n)$ be an integer sequence such that $b(n) = B$ after the whole transformation where we start with $A = n$, $B = 1$, $...
Notamathematician's user avatar
6 votes
1 answer
357 views

On A057985 and A287066

Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0 \to 01$, $1 \to 12$, $2 \to 0$). Let $b(n)$ be A287066 (i.e., start with $1$ and repeatedly substitute: $0 \to 01$, $1 \to 12$...
Notamathematician's user avatar
1 vote
1 answer
64 views

Sequence derived from transform of a given vector (with Fibonacci as partial sums)

Let F_n be A000045 (i.e., Fibonacci numbers). Here $$ F_n = F_{n-1} + F_{n-2}, \\ F_0 = 0, F_1 = 1 $$ Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). ...
Notamathematician's user avatar
2 votes
1 answer
111 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\...
Notamathematician's user avatar
2 votes
0 answers
54 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 $K$...
V. Asnin's user avatar
2 votes
2 answers
225 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 ...
Notamathematician's user avatar
1 vote
0 answers
69 views

Some ideas about parking functions and integer partitions

We know that a integer partition of $\lambda=(\lambda_1, ..., \lambda_m)$ of $n$ satisfying $\lambda_1\geq \cdots \geq \lambda_m$ and $\sum_{i=1}^m\lambda_i=n$. Let $\mathcal{P}(n)$ be the set of ...
Ethan's user avatar
  • 11
2 votes
1 answer
214 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. $abc$ ...
joro's user avatar
  • 24.3k
0 votes
0 answers
186 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 $$ \...
Notamathematician's user avatar
7 votes
1 answer
500 views

Suitable closed form for the A079501

Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position). The sequence begins with $$ 1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
Notamathematician's user avatar
2 votes
1 answer
107 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 ...
Notamathematician's user avatar
2 votes
0 answers
156 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 ...
Notamathematician's user avatar
4 votes
0 answers
115 views

Do all nonnegative integers appear in A051521?

For every positive integer $n$, $\tau(n)$ is the number of divisors of $n$. If we list the ratio of each positive integer $n$ to $\tau(n)$,they form a rational sequence 1,1,3/2,4/3,5/2,3/2,… Because $\...
Tong Lingling's user avatar
12 votes
4 answers
1k views

Six consecutive positive integers with certain shape

Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ? If they exist, one of those six integers A will be the product of 2 and a square of ...
Tong Lingling's user avatar
2 votes
0 answers
29 views

joint rank sequences

An algebraic question I have been working on led me to a sequence that appears in OEIS as A186355: "adjusted joint rank sequence of $(f(i))$ and $(g(j))$ with $f(i)$ before $g(j)$ when $f(i)=g(j)$...
Vladimir Dotsenko's user avatar
0 votes
0 answers
59 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 $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ ...
Notamathematician's user avatar
5 votes
0 answers
86 views

Formula and smallest solution for the A260711

Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$). The sequence begins with $$ 8, 16, 32, 48, 64, ...
Notamathematician's user avatar
1 vote
0 answers
63 views

On a numbers $k$ with specific $2$-adic valuation

Let $a(n)$ be A002326 (i.e., multiplicative order of $2 \operatorname{mod} 2n+1$). Let $b(n)$ be A179382 (i.e., the smallest period of pseudo-arithmetic progression with initial term $1$ and ...
Notamathematician's user avatar
0 votes
0 answers
95 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 $$ 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129 $$...
Notamathematician's user avatar
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&...
Notamathematician's user avatar
2 votes
1 answer
229 views

An integer sequence related to Pascal’s triangle

We need someone expert in binomial coefficients (subject 11B65) to recognize the integer sequence generated by an iterative formula we have encountered while working on a project about Pascal’s ...
Monk's user avatar
  • 125
5 votes
1 answer
195 views

Does every integer appear in the modular sum sequence?

$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by: $a(0) = 0, a(1) = 1$ and $a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
Dominic van der Zypen's user avatar
1 vote
0 answers
120 views

On a Fibonacci and binary

Let F(n) be A000045 i.e. Fibonacci numbers. Here $$ F(n) = F(n-1) + F(n-2), \\ F(0) = 0, F(1) = 1 $$ Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor $$ Let $$ T(n, k) = \left\lfloor\frac{n}{2^k}\...
Notamathematician's user avatar
1 vote
1 answer
135 views

Strongly regular binary sequences

Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A \subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty}\...
Dominic van der Zypen's user avatar
0 votes
0 answers
61 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 $...
joro's user avatar
  • 24.3k
2 votes
2 answers
192 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 ...
joro's user avatar
  • 24.3k
2 votes
1 answer
262 views

Curious sequences of polynomials

Given an integer $k\geq 2$, and $k+1$ invertible initial values $s_0,s_1,\ldots,s_k$ in some commutative ring $\mathcal A$ we set $$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{...
Roland Bacher's user avatar
2 votes
1 answer
99 views

Natural density of thickly syndetic set

A syndetic set $S$ is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural ...
Matej Moravik's user avatar
1 vote
1 answer
256 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\...
Ren Guan's user avatar
  • 101
0 votes
0 answers
53 views

Stolarsky representation from Zeckendorf representation with some pairs of bits inverted

Let $a(n)$ be A200714 i.e. Stolarsky representation interpreted as binary to decimal integers. Let $b(n)$ be A003714 i.e. Fibbinary numbers (Zeckendorf representation interpreted as binary to decimal ...
Notamathematician's user avatar
2 votes
1 answer
174 views

An upper bound on coefficients of some integer sequences

Given $\lambda>0$ let $B=B(\lambda)$ be the smallest integer such that there exist infinite integer sequences having values in $\lbrace 1,2,\ldots,B-1,B\rbrace$ and satisfying the following ...
Roland Bacher's user avatar
0 votes
0 answers
81 views

Partitions in A237981

Let $T(n,k)$ be A237981 i.e. array: row $n$ gives the NW partitions of n; see Comments. Here by $T(n,k)$ I mean $k$-th partition in $n$-th row. Let $$ \ell(n) = \left\lfloor\log_2 n\right\rfloor $$ ...
Notamathematician's user avatar
2 votes
2 answers
199 views

An identity for the ratio of two partial Bell polynomials

Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that the ...
qifeng618's user avatar
  • 942
1 vote
0 answers
108 views

Can the ideas of convex optimization be used to prove a bound?

If we define $\lambda(n)=\lfloor \log_2(n) \rfloor$ and $v(n)$ as the binary digit sum of positive integer $n$ we can make a toy example of what I think is the most important conjecture in addition ...
Neill Clift's user avatar
0 votes
0 answers
68 views

Recursions for the A111528

Let $T(n,k)$ be A111528 i.e. square table, read by antidiagonals, where the g.f. for row $n+1$ is generated by $$ xg_{n+1}(x) = \frac{1}{n+1}\left(1+nx - \frac{1}{g_n(x)}\right), \\ g_0(x) = \sum\...
Notamathematician's user avatar
3 votes
0 answers
116 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 ...
Notamathematician's user avatar
8 votes
4 answers
496 views

"Upside-down unimodal" sequences in combinatorics

Recall a sequence $a_0,\ldots,a_n$ of positive integers is unimodal if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in ...
Sam Hopkins's user avatar
13 votes
1 answer
667 views

When is $\mathrm{gcd}(k,p^k-1)=1$ true?

Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$? For the ...
Martin Brandenburg's user avatar
4 votes
0 answers
142 views

The smallest sequence without differences among Fibonacci numbers

Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$ of (strictly) positive integers, we can consider subsets $A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in $\mathcal S$. Examples: ...
Roland Bacher's user avatar
0 votes
1 answer
104 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 \...
Notamathematician's user avatar
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 $v(i)=m(i+1)-2$...
Notamathematician's user avatar
0 votes
0 answers
77 views

Constructing a pair of complementary sequences with the perfect differences

Let $F_n$ be A000045, i.e. Fibonacci numbers. Here $$ F_n = F_{n-1} + F_{n-2}, \\ F_0 = 0, F_1 = 1 $$ Let $g(n,m)$ be A257961. Here $$ g(n, m) = mF_{n-1} \operatorname{mod} F_n $$ Let $$ \varphi=\...
Notamathematician's user avatar
0 votes
0 answers
62 views

Simple non-recursive formula for inverse permutation to A316385

Let $$ \ell(n)=\left\lfloor\log_2 n\right\rfloor $$ Let $$ f(n)=n+2^{\ell(n)+1} $$ Let $a(n)$ be A316385, i.e. lexicographically earliest sequence of distinct positive terms such that for any $n > ...
Notamathematician's user avatar
3 votes
0 answers
68 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 $$ A(x)=\frac{1}{1-2x+\log(1-x)} $$ ...
Notamathematician's user avatar
3 votes
1 answer
225 views

Min problem on integers

Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that $$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
Nader Bshouty's user avatar

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