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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

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A006508:
a (n) = a (n  −  1) 
-th composite number [actually, nonprime (i.e. unit or composite) positive integer], 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, the formula for
a (n + 10)
is
     
a (n) 1 +
10
log a (n)
+
65
log 2 a (n)
+ O
1
log 3 a (n)
 .

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