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

(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.

• 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. – MathematicsStudent1122 Jul 5 at 4:49
• @MathematicsStudent1122, thank you for your comment. Please see this new question for the motivation. – Jose Arnaldo Bebita-Dris Jul 5 at 5:08
• Do you allow $n=1$? If so, then every integer greater than or equal to $2$ is one of these numbers, right? – Mark S. Jul 5 at 12:02
• @MarkS.: $n$ is any positive integer (i.e. it is not fixed). – Jose Arnaldo Bebita-Dris Jul 5 at 17:01

## 1 Answer

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".