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Template:Sequence of the Day for July 25

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Intended for: July 25, 2012

Timetable

  • First draft entered by Alonso del Arte on July 22, 2011 based on comments by Pab Ter and Nestor Romeral Andres. ✓
  • Draft reviewed by Daniel Forgues on July 25, 2011
  • Draft to be approved by June 25, 2012
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



A000058: Sylvester’s sequence, which gives the denominators
a (n)
for the greedy Egyptian representation of 1.
1 =
1
2
+
1
3
+
1
7
+
1
43
+
1
1807
+

(Of course the numerators are given by A000012.)

The denominators are given by the quadratic recurrence

     
a (1)  =  2;
a (n)  =  a (n − 1) [a (n − 1) − 1] + 1
 =  a (n − 1) 2a (n − 1) + 1,  n ≥ 2.

and by the formula (which shows that it is an infinite coprime sequence)

a (n) = 1 +
n  − 1
i  = 1
  
a (i),
where for
n = 1
we get 1 + (empty product, i.e. 1) = 2.