Template:Sequence of the Day for July 5

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

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

---

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 (  p q)  L

where

L

is Linnik’s constant. These give rise to bounds like

2 n

and

c  L n

, respectively.