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Topic:
stairway numbers (revised version)
Replies:
6
Last Post:
May 22, 2011 3:25 AM




Re: stairway numbers (revised version)
Posted:
May 17, 2011 2:27 AM


My problem was solved by several helpful members of sci.math:
'1x2 '3x4 '5x6 '7x8 '9x10 ... = ln2
'1x3 '5x7 '9x11 '13x15 '17x19 ... = pi/8
'1x4 '7x10 '13x16 ... = (ln2 + pi/sqrt3))/9 = 0.2785496...
It is then a composite of the first sums, and hardly a constant, as I had hoped.
Reconstructing early and very early methods is a part of my profession, playing with numbers a hobby of mine. Here is a quick approximation of Euler's gamma found yesterday:
1 minus ln(1) minus '7 of 3 1 '2 minus ln(2) minus '13 of 3 1 '2 '3 minus ln(3) minus '19 of 3 1 '2 '3 '4 minus ln(4) minus '25 of 3
Gauss connected the natural logarithm with pi, while Euler combined pi and the primes in the infinite product 3/2 x 5/6 x 7/4 x 11/10 x 13/14 x 17/18 x 19/18 x 23/22... approximating pi/2, containing the odd primes and the neighboring numbers that don't contain 4 as factor.
I shall go on playing with number patterns, fascinated by how quickly the natural numbers lead to the natural logarithm, and the odd numbers to pi.



