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Template:Sequence of the Day for February 13

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Intended for: February 13, 2012

Timetable

  • First draft entered by Alonso del Arte on November 13, 2011
  • Draft reviewed by Daniel Forgues on February 02, 2017
  • Draft to be approved by January 13, 2012

Yesterday's SOTD * Tomorrow's SOTD

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



A066340: Fermat’s triangle:
T  (n, m) = m  φ (n) mod n
, for
n   ≥   2
and
1   ≤   m   ≤   n  −  1
. (Concatenated rows of Fermat’s triangle.)
{1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, ... }

Fermat’s triangle
n
       
n  − 1

k  = 1
T  (n, k )

2   1  
1
3   1 1  
2
4   1 0 1  
2
5   1 1 1 1  
4
6   1 4 3 4 1  
13
7 1 1 1 1 1 1  
6
8   1 0 1 0 1 0 1  
4
9   1 1 0 1 1 0 1 1  
6
10   1 6 1 6 5 6 1 6 1  
33
11   1 1 1 1 1 1 1 1 1 1  
10
12 1 4 9 4 1 0 1 4 9 4 1  
38
13   1 1 1 1 1 1 1 1 1 1 1 1  
12
14   1 8 1 8 1 8 7 8 1 8 1 8 1  
61
15   1 1 6 1 10 6 1 1 6 10 1 6 1 1  
52
16 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  
8

k = 1

2
3
4
5
6
7
8
9
10
11
12
13
14
15  

Per Fermat’s little theorem, the rows corresponding to prime
n
consist of all 1s, although the converse is not true: rows corresponding to composite
n
which consist of all 1s are the Carmichael numbers. Also, the prime powers consist only of ones and zeros, e.g.
  • powers of 2: 1, 0 (repeated pattern);
  • powers of 3: 1, 1, 0 (repeated pattern);
  • ...

Is the converse also false? That is, are there “pseudoprimepowers” to all bases, i.e. giving a repeated pattern identical to some prime power?

A?????? Row sums of Fermat’s triangle:
n   −  1

m  = 1
 (m  φ (n) mod n)
, for
n   ≥   2
.
{1, 2, 2, 4, 13, 6, 4, 6, 33, 10, 38, 12, 61, 52, 8, ...}