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For any $n\geq 2$ consider the recursion \begin{align*} a(0,n)&=n;\\ a(m,n)&=a(m-1,n)+\operatorname{gcd}(a(m-1,n),n-m),\qquad m\geq 1. \end{align*} I conjecture that $a(n-1,n)$ is always prime. To ve …

Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here $$a(n)=\frac{1}{n+1}\binom{2n} …

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $ …

Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns). Let $b(n)$ be an integer sequence with generating function $B(x)$ such that $$ B(x) = \cf …

Let $a(n)$ be A261041 (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts). Let $b(n)$ be an integer sequence with generatin …

Please note that this question has been completely reworked in order not to overload it with unnecessary and useless information. Let $f(n)$ be an arbitrary function with integer values. Let $a(n)$ b …

Let $a(n)$ be A204262 i.e. permanent of the matrix $n\times n$ with elements $\min(i,j)$. Let $$ f_{n,\ell}(x)=g_{n,\ell}(x)+f_{n,\ell-1}(\ell)-g_{n,\ell}(\ell), \\ g_{n,\ell}(x)=\int (n-\ell)^2 f_{n …

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, …

Let $a(n)$ be A214973, number of terms in greedy representation of $n$ using Fibonacci and Lucas numbers. Let $b(n)$ be A329320, sequence which arises from attempts to simplify computing of A329319. H …

Let $\eta(n)$ be A006337, an "eta-sequence" defined as follows: $$\eta(n)=\left\lfloor(n+1)\sqrt{2}\right\rfloor-\left\lfloor n\sqrt{2}\right\rfloor$$ Sequence begins $$1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1 …

Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$. The sequence begins with $$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704 …

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 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 $A = n$, $B = 1$, …

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 permutat …

Let $a(n)$ be A309099 i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$. We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrer …