Is there such a equivalence ralation existing on a set of natural number N?

Question

The corresponding quotient(or the set of equivalence classes) of this relation is

{ {1}, {2,3}, {4,5,6}, {7,8,9,10}, {11,12,13,14,15},...}

We thought of lots of answers at today's class but no one is correct...I am confused now... Is there such a relation existing on set N?

Answer 1

Sure. Consider this relation: $m\sim n$ when one of these conditions occur:

• $m,n\in\{1\}$;
• $m,n\in\{2,3\}$;
• $m,n\in\{4,5,6\}$;
• $m,n\in\{7,8,9,10\}$;
• $\vdots$

Answer 2

Your numbers are equivalent when they have the same least triangle number strictly greater than them. Since$$T_{k-1}\le n<T_k\iff(2k-1)^2\le 8n<(2k+1)^2\iff k-1\le\frac{\sqrt{8n}-1}{2}<k,$$your relation has $m\sim n$ iff $\left\lfloor\frac{\sqrt{8m}-1}{2}\right\rfloor=\left\lfloor\frac{\sqrt{8n}-1}{2}\right\rfloor$.