# Questions tagged [integer-sequences]

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

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### 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 $f(n)$ (...
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### 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 ...
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### 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 ...
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### 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 $K$...
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### 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 ...
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### 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 $$3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129$$...
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### 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&...
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### 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 ...
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### 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 ...
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### 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 ...
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### "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 ...
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### 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 ...
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### 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: ...
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