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Self-numbers and generators (Posted on 2018-05-23) Difficulty: 4 of 5
begin background
For any positive integer n, define d(n) to be n plus the sum of the digits of n.
For example, d(79) = 79 + 7 + 9 = 95.
Take integer n as a starting point, you can construct the infinite increasing sequence of integers n, d(n), d(d(n)), d(d(d(n))), ....etc
For example, if you start with 33, the next number is 33 + 3 + 3 = 39, the next
is 39 + 3 + 9 = 51, the next is 51 + 5 + 1 = 57, and so you generate the sequence : 33, 39, 51, 57, 69, 84, 96, 111, 114, 120, 123, 129, 141, ...
The number n is called a generator of d(n). In the sequence above, 33 is a generator of 39, 39 is a generator of 51, 51 is a generator of 57, and so on. Some numbers have more than one generator: for example, 101 has two generators, 91 and 100.
A number with no generators is a self-number.

There are thirteen self-numbers less than 100: 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, and 97.

end background

Now come my questions:

a.(d2) What is the smallest n making a 10n a self number?

b.(d4) Checking integers with 1 in the beginning, 1 in the end and n-1 zeros between the ones (i.e.10n+1) what value of n creates a number with 3 generators ?

c.(d5 or d4 after a hint) What is the smallest number with 4 generators ?

No Solution Yet Submitted by Ady TZIDON    
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Comments: ( Back to comment list | You must be logged in to post comments.)
re(3): Thoughts on c. (assist req'd) ====>Hint Comment 8 of 8 |
(In reply to re(2): Thoughts on c. (assist req'd) ====>Hint by Jer)

You are  right.


However no harm here since none of us has the means to reach that far.
Hope that we were exposed to something new and educational.

Edited on May 27, 2018, 3:15 am
  Posted by Ady TZIDON on 2018-05-27 02:51:01

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