Define $\triangle_n$ to be the $n$th triangular number.

Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$

Define $(\ell,k)$-smough numbers to be numbers that have all prime divisors $p$ of the form of either $p<\ell$ or $k<p$.

1. Is there an $\ell_0\geq6$ such that for every $\ell>\ell_0$ and for every $k$, there are an infinite number of $k$ for which there is always an $n$ such that $M_n$ is $(\ell,k)$-smough?

2. Given an $\ell$ and a $k$, how small can $n$ be such that $M_n$ is $(\ell,k)$-smough?

Define $\triangle_n$ to be the $n$th triangular number.

Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$

Define $(\ell,k)$-smough numbers to be numbers that have all prime divisors $p$ of the form of either $p<\ell$ or $k<p$.

1. Is there an $\ell_0\geq6$ such that for every $\ell>\ell_0$ there is a $k_\ell$ such that for every $k\geq k_\ell$ there is always an $n$ such that $M_n$ is $(\ell,k)$-smough?

2. Given an $\ell$ and a $k$, how small can $n$ be such that $M_n$ is $(\ell,k)$-smough?

Define $\triangle_n$ to be the $n$th triangular number.

Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(2\triangle_n^2-1).$$

Define $(\ell,k)$-smough numbers to be numbers that have all prime divisors $p$ of form either $p<\ell$ or $k<p$.

1. Is there an $\ell_0\geq6$ such that for every $\ell>\ell_0$ and for every $k$ are there infinite number of $k$ for which there is always an $n$ such that $M_n$ is $(\ell,k)$-smough?

2. Given an $\ell$ and a $k$ how small can $n$ be such that $M_n$ is $(\ell,k)$-smough?

Define $\triangle_n$ to be the $n$th triangular number.

Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$

Define $(\ell,k)$-smough numbers to be numbers that have all prime divisors $p$ of the form of either $p<\ell$ or $k<p$.

1. Is there an $\ell_0\geq6$ such that for every $\ell>\ell_0$ and for every $k$, there are an infinite number of $k$ for which there is always an $n$ such that $M_n$ is $(\ell,k)$-smough?

2. Given an $\ell$ and a $k$, how small can $n$ be such that $M_n$ is $(\ell,k)$-smough?

Define $(\ell,k)$-smough numbers to be numbers that have prime divisors $p$ of form either $p|\ell!$ or $k<p$.

1. Given an $\ell$ is there a maximum $k$ above which there is no $n$ such that $q(n(n+1))$ is $(\ell,k)$-smough where $q(x)=x(x^2-1)$?

2. Is there a fast algorithm to find $n$ for a given $\ell,k$ such that $q(n(n+1))$ is $(\ell,k)$-smough when such $n$ exists or return $n=\infty$?

Note $q(n(n+1))$ is always of form $2\triangle_n(2^2\square_n-1)$ where $\triangle_n$ is $n$-th triangular number (sum of first $n$ natural numbers) and $\square_n=\triangle_n^2$ is sum of cubes of first $n$ natural numbers.

How large can the ratio $\frac k\ell$ be?

Interesting cases at $n<100$ are at $17,31,59,89,97$:

https://www.wolframalpha.com/input/?i=Factor%28%28p%29%28p%2B1%29%28%28%28p%29%28p%2B1%29%29%5E2-1%29%29+at+p%3D17

https://www.wolframalpha.com/input/?i=Factor%28%28p%29%28p%2B1%29%28%28%28p%29%28p%2B1%29%29%5E2-1%29%29+at+p%3D31

https://www.wolframalpha.com/input/?i=Factor%28%28p%29%28p%2B1%29%28%28%28p%29%28p%2B1%29%29%5E2-1%29%29+at+p%3D59

https://www.wolframalpha.com/input/?i=Factor%28%28p%29%28p%2B1%29%28%28%28p%29%28p%2B1%29%29%5E2-1%29%29+at+p%3D89

https://www.wolframalpha.com/input/?i=Factor%28%28p%29%28p%2B1%29%28%28%28p%29%28p%2B1%29%29%5E2-1%29%29+at+p%3D97

I doubt $k$ grows much if $\ell$ is fixed.

Define $\triangle_n$ to be the $n$th triangular number.

Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(2\triangle_n^2-1).$$

Define $(\ell,k)$-smough numbers to be numbers that have all prime divisors $p$ of form either $p<\ell$ or $k<p$.

1. Is there an $\ell_0\geq6$ such that for every $\ell>\ell_0$ and for every $k$ are there infinite number of $k$ for which there is always an $n$ such that $M_n$ is $(\ell,k)$-smough?

2. Given an $\ell$ and a $k$ how small can $n$ be such that $M_n$ is $(\ell,k)$-smough?