3 problems

For integers $r\geq2$ and $n\geq3$, let $s_r^*(n)$ be the smallest positive integer, if it exists, such that every integer $s\geq s_r^*(n)$ has at least $r$ different $n$-partition…

For integers $r\geq2$ and $n\geq3$, let $s_r^*(n)$ be the smallest positive integer, if it exists, such that every integer $s\geq s_r^*(n)$ has at least $r$ different $n$-partition…

Let $X=\\{x_1,x_2,x_3\\}$ be a 3-partition, and write $s=x_1+x_2+x_3$ and $p=x_1x_2x_3$. Define $T\\{x_1,x_2,x_3\\}=(s,p,3)$, and call a triple $(s,p,3)$ admissible if at least two…