[seqfan] A205827 and A111870

[John W. Nicholson]

The title of A205827 is "The smallest prime(n) such that (prime(n+1)/prime(n))^n is increasing." I am wondering if it should be "The smallest prime(k) such that (prime(k+1)/prime(k))^n is increasing."? An example would help to make it clear too. Also, if the above is a correction, then the value of k should be a related sequence, correct?


I take the reason why the sequence is limited to a(25) is because no one has check beyond 10^12, yet the comment implies A111870 which has been extended to a(32). Should/can A205827 be extended using A111870 values to a(33)?

Also, I added "a(4) = " to Farideh Firoozbakht's, May 07 2012 comment as to make it clear. 

While talking about A111870, I see the comment "Comment from Jose Brox, Dec 31, 2005: As I understand it, the sequence refers to "Smallest prime p such that its following gap has bigger merit than the other primes smaller than p." If that is the case, then it has an error. The sequence starts: 2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 4652353, 2010733... but you can see that 4652353 > 2010733, so in any case it should be listed after, not before it. But above that, its merit is 10.03 < 10.20, the merit of 2010733, so it is not in a mistaken position: it shouldn't appear on the sequence." Should this still be in the comments? I know that it is historical, however I do not see the value of leaving it visible to everyone's first look at the sequence. I think removing it and making a comment with the history is enough.

 

John W. Nicholson