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

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Franz Gnaedinger

Posts: 330
Registered: 4/30/07
Re: stairway numbers (revised version)
Posted: May 17, 2011 2:27 AM
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My problem was solved by several helpful members of

'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

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.

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