conjecture

Mar 2019
318
14
iran
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
 
Aug 2012
2,495
781
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?
 
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Mar 2019
318
14
iran
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).
 
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