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# Template:Sequence of the Day for October 25

Intended for: October 25, 2012

## Timetable

• First draft entered by Alonso del Arte based on comments by Robert G. Wilson v on October 23, 2011
• Draft to be reviewed by August 25, 2012
• Draft to be approved by September 25, 2012

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A006508: a(n) = a(n − 1)-th composite number, with a(0) = 1.

 { 1, 4, 9, 16, 26, 39, 56, 78, ... }

This sequence is generated by a sieve: start with the natural numbers, remove those terms which occupy positions which are prime, leaving {1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...}; remove those terms whose positions are primes plus one; leaving {1, 4, 9, 12, 15, 16, 18, ...}; remove those whose positions are primes plus two; and so on...

What is the asymptotic behavior of this sequence? The Bojarincev asymptotic formula for the composite numbers allows a formula for a(n + k) for any fixed k in terms of a(n). For example,

$a(n+10) = a(n)\left(1+\frac{10}{\log a(n)}+\frac{65}{\log^2 a(n)}+O\left(\frac{1}{\log^3 a(n)}\right)\right). \,$

But is there a reasonable asymptotic for a(n) without using earlier values?