, where is the list of the digits of an integer n.

,

and hence we have {134, 284, 16777476, 3387741, 18424613, 16824417 }.

When the number of 9 in the list is exactly 9,then . .

Suppose that the number of 9 in the list is exactly 8. The remainning 2 numbers are =3 and an integer such that .

When , then .

When , then .

When the number of 9 in the list is less than 8, we have .

we have .

. When , by Theorem 3 . If , by Theorem 3 we have . In this way this sequence decreases while the value of the sequence is bigger or equal to 3000000000. Therefore there exists a natural number t such that . In this way we can prove that there are infinite number of m such that , and hence there is a non-negative integer u such that for at least two values of m. Let v be the smallest integer such that and let w be the second smallest integer such that .

Then clearly

,..., is a loop.

{132,32,31,28,16777220,2517295,388250291,420978083,421798753, 405848204,33558331,16783575,18477236,18471246, 17647933,389114542,404201375,827214,17601024,870463, 17647699,776581633,18520794,405024635,53451,6534,50064, 50039,387423643,17648012,17647678,19341414,387421287, 35201810,16780376,18517643,17650825,17653671,1743552,830081, 33554462,53476,873607,18470986,421845378,34381644,16824695, 404294403,387421546,17651084,17650799,776537847,20121452, 3396,387467199,793312220,388244100,33554978,405027808, 34381363,16824237,17647707,3341086,16824184,33601606,140025, 3388,33554486,16830688,50424989,791621836,405114593, 387427281,35201810,16780376,18517643,17650825,17653671, 1743552,830081,33554462,53476,873607,18470986,421845378, 34381644,16824695,404294403,387421546,17651084,17650799, 776537847,20121452,3396,387467199,793312220,388244100, 33554978,405027808,34381363,16824237,17647707,3341086, 16824184,33601606,140025,3388,33554486,16830688,50424989, 791621836,405114593,387427281,35201810,16780376}.

*From this sequence we get Graph 3.1.
*

You will also find the same number appears twice in the sequence if you look at the sequence carefully.