Questions about A389140 and A389141

[Ali Sada]

Hi everyone,

 

Hope all is well. I submitted these two sequences and I have some questions. I would really appreciate your answers.

 

1.      A389140: “Smallest final number reachable by successively dividing a substring of digits of n by 2.”

(Michael clarified: The substring replaced must be an even number and cannot begin with a 0 digit. There may be multiple different replacements possible and the aim is whatever steps reach the smallest final value.)

 

Q1: Is it worth adding a sequence for the numbers that terminate to 1?

1,2,4,8,16,26,32,34,38,46,52,54,58,64,68,…

             

Q2: Even numbers that are not multiples of 5 always terminate to 1 or 3, while even multiples of 5 terminate to 5,15,35,... but not all multiples of 5 with odd digits appear. What’s the rule here?

 

2.      A389141: “Least positive integer k such that A389140 (k*n) has the smallest value.”

It seems that if n is not a multiple of 5, there is always some multiplier k that makes A389140(k*n) collapse to 1.

Pontus added “It seems that the minimum of A389140(k*n) is 15 if n is divisible by 25, 5 if n is divisible by 5 (but not by 25), and 1 otherwise.”

Q3: Can we prove this?

Q4: If we have the proof, shouId I change the definition to “Least positive integer k such that A389140 (k*n) is 1, 5, or 15."? 

Best,

 

Ali