April 4th, 2019, 10:15 AM  #1 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12  conjecture
A(n) = m if 1/1 + 1/2 + 1/3 + ... + 1/(m1) < n < 1 + 1/2 + 1/3 + ... + 1/(m) for example A(1) = 1 A(2) = 4 1/1 + 1/2 + 1/3 < 2 < 1/1 + 1/2 + 1/3 + 1/4 A(3) = 11 A(4) = 31 A(5) = 83 A(6) = 227 A(7) = 614 conjecture lim A(n+1)/A(n) = e for example A(2)/A(1) = 4 A(3)/A(2) = 2.75 A(4)/A(3) = 2.82 A(5)/A(4) = 2.68 A(6)/A(5) = 2.73 A(7)/A(6) = 2.70 
April 4th, 2019, 12:01 PM  #2 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12 
apparently this has already been discovered A004080  OEIS 
April 5th, 2019, 05:39 PM  #3  
Senior Member Joined: Aug 2008 From: Blacksburg VA USA Posts: 346 Thanks: 6 Math Focus: primes of course  Quote:
 
April 5th, 2019, 08:56 PM  #4 
Senior Member Joined: Aug 2012 Posts: 2,321 Thanks: 714 
Yes that's a nice insight. Youngmath do you attempt proofs for your ideas? Or do you do all this by intuition and just seeing it?
Last edited by Maschke; April 5th, 2019 at 08:59 PM. 
April 5th, 2019, 10:11 PM  #5 
Senior Member Joined: Mar 2019 From: iran Posts: 308 Thanks: 12 
Of course I have proofs for them. I got a formula for number of twin primes less than a given number that works well up to 100000, but for larger numbers that has an error and the error gets bigger. It's proof was like my proof for pi(n).
Last edited by skipjack; April 6th, 2019 at 12:48 AM. 

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