Template:Sequence of the Day for February 13

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.

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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, ...}