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Template:Sequence of the Day for July 3

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Intended for: July 3, 2012

Timetable

  • First draft entered by Alonso del Arte on March 5, 2011 based on an almost verbatim copy of a write-up by David W. Wilson from October 19, 2010. ✓
  • Draft reviewed by Daniel Forgues on April 30, 2011
  • Draft approved by Charles R Greathouse IV on July 1, 2011
Yesterday's SOTD * Tomorrow's SOTD

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



A020639
lpf (n)
: least prime dividing
n
(
a (1) = 1
).
{ 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, ... }

I’m listing the elements without the initial 1. I’m a purist, and given that 1 has no prime factors, I feel a(1) is undefined and should not appear in the sequence. I’m sure I (David W. Wilson) did not include it when I authored the sequence.

Why do I find this sequence SOTD-worthy? It is because of a question I posed back in 1997. I noticed that in this sequence, between every pair of 2's, there was an element > 2, easy enough to prove. Similarly, I could show that between every pair of 3's there is an element > 3. I found the same to be true of 5, 7, 11, and 13. I began to think it might true for every prime number, but I could not find a general proof.

So I posed my question the seqfan list. Apparently it was interesting, because it sent John Conway, Johan de Jong, Derek Smith, and Manjul Bhargava to the blackboard find a counterexample. After two hours, they found one:

n = 126972592296404970720882679404584182254788131, with a(n) = a(n+226) = 113 and all intervening values < 113.

Later, Fred Helenius found that the smallest prime that generates a counterexample is 71, for which

n = 7310131732015251470110369, with a(n) = a(n+142) = 71, and all intervening values < 71.

He also found the earliest known counterexample,

n = 2061519317176132799110061, with a(n) = a(n+146) = 73, and all intervening values < 73.

I was rather amazed that you had to look so far into the sequence.