Template:Sequence of the Day for July 25

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

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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) 2 − a (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.