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April 4th, 2019, 10:15 AM   #1
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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
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April 4th, 2019, 12:01 PM   #2
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apparently this has already been discovered
A004080 - OEIS
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April 5th, 2019, 05:39 PM   #3
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Math Focus: primes of course
Quote:
Originally Posted by youngmath View Post
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.
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April 5th, 2019, 08:56 PM   #4
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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.
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April 5th, 2019, 10:11 PM   #5
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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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