Template:Sequence of the Day for October 25

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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

Yesterday's SOTD * Tomorrow's SOTD

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

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?

Template:Sequence of the Day for October 25

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