There is no mathematical way to easily determine from which number onwards such a sequence covers all numbers. The game Sieve give us a tool to solve such usually tricky problems easily and quickly. The game Sieve can also be used to create a prime number sieve (see variants).

Click the columns of digits at the left border to set a starting number (number '8' in the above example).
Click the STEP range to set the increment (number '3' in the above equation).
Stepsize=0 will cause the system to draw the starting number only, which is useful for marking single numbers on the grid.

Click the colour selection to set the colour of the sequence.
To especially mark the first grid entry of every sequence, click the coloured box next to 'First:'.
Hit the RUN button to start the automatic drawing of the sequence.

A special function will be triggered when you click the 'Start at no' button. The text of this button will change to 'use all'.
In this mode the selected sequence (3*n in our example) will be added to each number on the grid! The selected start number will be ignored in this case.

Application:
The 'use all' mode allows you for example to find out from which natural number k onwards the formula 8 + 13*n + 15*m + 27*p + 41*q covers all natural numbers. Without this Sieve game, one would have to write a separate program for each such search. Sieve allows you to solve any such problem in a few seconds!

The default board is 50x50. There are also variants with boards 10x10 and 100x100.
Several variants with examples have been added to help you understand this tool.
For details see the associated game texts.

 
To my knowledge, Sieve is the first program ever to give offer a graphical user interface for sieve-problems.