Answer
There are infinitely many numbers that satisfy the characteristics of the given problem.
Work Step by Step
Let $x$ be the integer. Then the next 2 consecutive integers are $x+1$ and $x+2$. Based on the conditions of the problem, then,
\begin{array}{l}\require{cancel}
x+(x+1)+(x+2)=3(x+1)
\\\\
(x+x+x)+(1+2)=3x+3
\\\\
3x+3=3x+3
\\\\
3x-3x=3-3
\\\\
0=0 \text{ (TRUE)}
.\end{array}
Since the solution above ended with a TRUE statement, the given equation is an identity. This means that there are infinitely many consecutive numbers that satisfy the given equation. In fact, any three consecutive integers will follow the characteristic that the sum of $3$ consecutive integers is three times the second integer.
