My Math Forum conjecture

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 April 4th, 2019, 10:15 AM #1 Senior Member   Joined: Mar 2019 From: iran Posts: 318 Thanks: 14 conjecture A(n) = m if 1/1 + 1/2 + 1/3 + ... + 1/(m-1) < 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: 318 Thanks: 14 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: 354
Thanks: 7

Math Focus: primes of course
Quote:
 Originally Posted by youngmath apparently this has already been discovered A004080 - OEIS
There is no shame in original rediscovery, and it can be thought of as encouraging in my mind.

 April 5th, 2019, 08:56 PM #4 Senior Member   Joined: Aug 2012 Posts: 2,414 Thanks: 755 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: 318 Thanks: 14 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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