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

 Messages: [ Previous | Next ]
 Franz Gnaedinger Posts: 330 Registered: 4/30/07
stairway numbers (revised version)
Posted: Jan 22, 2010 1:51 AM

Developing the Horus eye series

1 = '1
1 = '2 '2
1 = '2 '4 '4
1 = '2 '4 '8 '8
1 = '2 '4 '8 '16 '16
1 = '2 '4 '8 '16 '32 '32
1 = '2 '4 '8 '16 '32 '64 '64

Developing an alternativer series

1 = '1
1 = '1x2 '2
1 = '1x2 '2x3 '3
1 = '1x2 '2x3 '3x4 '4
1 = '1x2 '2x3 '3x4 '4x5 '5
1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6
1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7

Infinite series

'1x2 '2x3 '3x4 '4x5 '5x6 '6x7 ... = 1

'1x3 '3x5 '5x7 '7x9 '9x11 '11x13 ... = '2

'1x4 '4x7 '7x10 '10x13 '13x16 '16x19 ... = '3

Partial series

'1x2 '3x4 '5x6 ... = ln2

'1x3 '5x7 '9x11 ... = pi/8

'1x4 '7x10 '13x16 ... = ???

Complementary stairways approximating ln2

'1x2 '2x2
'1x2 '3x4 '4x2
'1x2 '3x4 '5x6 '6x2
'1x2 '3x4 '5x6 '7x8 '8x2

1 minus '2x3 '3x2
1 minus '2x3 '4x5 '5x2
1 minus '2x3 '4x5 '6x7 '7x2
1 minus '2x3 '4x5 '6x7 '8x9 '9x2

Complementary stairways approximating pi/8

'1x3 '4x4
'1x3 '5x7 '8x4
'1x3 '5x7 '9x11 '12x4
'1x3 '5x7 '9x11 '13x15 '16x4

'2 minus '3x5 '6x4
'2 minus '3x5 '7x9 '10x4
'2 minus '3x5 '7x9 '11x13 '14x4
'2 minus '3x5 '7x9 '11x13 '15x17 '18x4

Complementary stairways approximating ???

'1x4 '6x6
'1x4 '7x10 '12x6
'1x4 '7x10 '13x16 '18x6
'1x4 '7x10 '13x16 '19x22 '24x6

'3 minus '4x7 '9x6
'3 minus '4x7 '10x13 '15x6
'3 minus '4x7 '10x13 '16x19 '21x6
'3 minus '4x7 '10x13 '16x19 '22x25 '27x6

I am still wondering about the last number.
Does anybody recognize it? My hope is that
it might perhaps play a role in the theory
of prime numbers.

Date Subject Author
1/22/10 Franz Gnaedinger
4/6/10 Diesellin
5/14/11 Franz Gnaedinger
5/16/11 Franz Gnaedinger
5/17/11 Franz Gnaedinger
5/19/11 Franz Gnaedinger
5/22/11 Franz Gnaedinger