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

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?