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We define $$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2} \sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$ (Searching the OEIS yielded no results.) For $n>2$ we have the following experimental observations (checked …

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_{ …

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 property: …

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: I …

Sequence A76132 starting as $1,1,2,4,10,36,218,\ldots$ of the OEIS is recursively defined by $a(1)=1$ and $a(n)=\sum_{k=1}^{n-1}a(n-k)^k$ for $n\geq 2$. It is eventually periodic of period 1,1 and 34 …

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 …