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Template:Sequence of the Day for July 5

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Intended for: July 5, 2012

Timetable

  • First draft entered by Alonso del Arte on March 6, 2011 based on an almost verbatim copy of a write-up by Charles Greathouse from October 20, 2010. ✓
  • Draft reviewed by Alonso del Arte on April 30, 2011
  • Draft approved by T. D. Noe June 5, 2011
Yesterday's SOTD * Tomorrow's SOTD

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A177854: Smallest prime of rank
n
.
{ 2, 3, 11, 131, 1571, 43717, 5032843, 1047774137, ... }
This is a re-imagination of the Erdős–Selfridge classification of primes. 2 is rank 0 by definition, and for other primes
p
the rank is the lesser of the maximum rank of the primes dividing
p  −  1
and the maximum rank of the primes dividing
p  +  1
. There are many interesting questions that could be asked about this sequence. What are
a (8)
,
a (9)
, and
a (10)
? The sequence is infinite: how closely can its growth be bounded? (Look at the logarithmic graph, it seems to follow quite close to a regular pattern.) How does the closely-related sequence A169818 behave around random large integers? (This last question is related to the performance of modern
n  −  1 / n  +  1 / n 2  −  1
algorithms!) Trivial bounds: If
p
and
q
are the smallest primes of rank
r  −  1
, then the smallest prime of rank
r
is at least
2 p  +  1
and at most
c (  pq)  L
where
L
is Linnik’s constant. These give rise to bounds like
2n
and
c  Ln
, respectively.