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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?

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  • $\begingroup$ Out of curiosity, is the $\triangle$ and $\square$ notation standard? $\endgroup$
    – user147820
    Commented Aug 14, 2020 at 4:50
  • $\begingroup$ $\triangle$ is standard for triangular numbers. $\square$ represents sum of cubes (three dimensional squares). $\endgroup$
    – VS.
    Commented Aug 14, 2020 at 4:56
  • $\begingroup$ Right right sorry I read in the post, was just wondering if that notation is commonly used $\endgroup$
    – user147820
    Commented Aug 14, 2020 at 4:58
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    $\begingroup$ $\triangle$ was used by Gauss. $\endgroup$
    – VS.
    Commented Aug 14, 2020 at 5:02

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