(Note: Please see this new question for the motivation.)

A number $T$ is said to be triangular if it could be written in the form $$T=\frac{n(n+1)}{2},$$ where $n$ is a positive integer.

Here is my question:

Is there a specific terminology for numbers which are nontrivial multiples of triangular numbers?

That is, is there a specific name for numbers $T'$ which are of the form $$T' = dT = d\cdot\bigg(\frac{n(n+1)}{2}\bigg),$$ for some integer $d > 1$?

I tried searching for the relevant sequence in OEIS, but all I am seeing are references to $T'$ being triangular as well.

  • 1
    $\begingroup$ Is there any motivation to this question? You can take any known sequence $a_n$ of positive integers and consider $S = \{k \in \mathbb{N} : k = d \cdot a_j \ \text{for} \ d \in \mathbb{N}_{>1}, j \in \mathbb{N}\}$. In general, this set won't be well known, or even have a name. $\endgroup$ – MathematicsStudent1122 Jul 5 at 4:49
  • $\begingroup$ @MathematicsStudent1122, thank you for your comment. Please see this new question for the motivation. $\endgroup$ – Jose Arnaldo Bebita-Dris Jul 5 at 5:08
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    $\begingroup$ Do you allow $n=1$? If so, then every integer greater than or equal to $2$ is one of these numbers, right? $\endgroup$ – Mark S. Jul 5 at 12:02
  • $\begingroup$ @MarkS.: $n$ is any positive integer (i.e. it is not fixed). $\endgroup$ – Jose Arnaldo Bebita-Dris Jul 5 at 17:01

I'm going to answer based on the confirmation in the comment.

  • Note that if $T'>1$, then taking $d=T'$ and $n=1$ (so that $T=1*2/2=1$) works.
  • Also, since $d>1$, $T'=1$ is impossible.
  • And since $n,d\ge1$, $T'<1$ is impossible as well.

Therefore, a name for the possible values of $T'$ is "integers greater than 1".


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