Sequences the same except for 1 term?

[Hans Havermann]

Leroy Quet wrote:

>> As some of you may have observed, the sequences are the same for  
>> the given terms except for the 5th term, which is 14 for A113118  
>> and is 15 for A113117.
>>
>> Question: Is the 5th term the only differing term among both  
>> sequences?

Possible outline for a proof that it is the only term:

1. Show that for terms > 7, the second-highest prime <= a(n-1) is the  
only one in contention for having a smallest multiple > a(n-1) that  
might be bigger than the smallest multiple > a(n-1) of the highest  
prime <= a(n-1). [I believe this is true for the first 30 terms.]

2. Show that for terms > 5, this second-highest prime does not  
produce such a situation. [I believe this is true for the first 1000  
terms.]
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