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Template:Sequence of the Day for July 3
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 ✓
The line below marks the end of the <noinclude> ... </noinclude> section.
| lpf (n) |
| 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.
