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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.



