Coprimality triangle

Note, among others, the patterns highlighted in blue and green (symmetric about multiples of 

6

) of the noncoprime numbers.

The equilateral version of the coprimality triangle
(Coprime numbers triangle)

n																																																																		$\varphi (n)=\,$ $\sum _{i=1}^{n-1}=T(n,i)\,$
2																																	1																																	1
3																																1		1																																2
4																															1		0		1																															2
5																														1		1		1		1																														4
6																													1		0		0		0		1																													2
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		0		1		0		0		0		1		0		1																									4
11																								1		1		1		1		1		1		1		1		1		1																								10
12																							1		0		0		0		1		0		1		0		0		0		1																							4
13																						1		1		1		1		1		1		1		1		1		1		1		1																						12
14																					1		0		1		0		1		0		0		0		1		0		1		0		1																					6
15																				1		1		0		1		0		0		1		1		0		0		1		0		1		1																				8
16																			1		0		1		0		1		0		1		0		1		0		1		0		1		0		1																			8
17																		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1																		16
18																	1		0		0		0		1		0		1		0		0		0		1		0		1		0		0		0		1																	6
19																1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1																18
20															1		0		1		0		0		0		1		0		1		0		1		0		1		0		0		0		1		0		1															8
21														1		1		0		1		1		0		0		1		0		1		1		0		1		0		0		1		1		0		1		1														12
22													1		0		1		0		1		0		1		0		1		0		0		0		1		0		1		0		1		0		1		0		1													10
23												1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1												22
24											1		0		0		0		1		0		1		0		0		0		1		0		1		0		0		0		1		0		1		0		0		0		1											8
25										1		1		1		1		0		1		1		1		1		0		1		1		1		1		0		1		1		1		1		0		1		1		1		1										20
26									1		0		1		0		1		0		1		0		1		0		1		0		0		0		1		0		1		0		1		0		1		0		1		0		1									12
27								1		1		0		1		1		0		1		1		0		1		1		0		1		1		0		1		1		0		1		1		0		1		1		0		1		1								18
28							1		0		1		0		1		0		0		0		1		0		1		0		1		0		1		0		1		0		1		0		0		0		1		0		1		0		1							12
29						1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1						28
30					1		0		0		0		0		0		1		0		0		0		1		0		1		0		0		0		1		0		1		0		0		0		1		0		0		0		0		0		1					8
31				1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1				30
32			1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1		0		1			16
			i = 1		2		3		4		5		6		7		8		9		10		11		12		13		14		15		16		17		18		19		20		21		22		23		24		25		26		27		28		29		30		31

Definition

The entries of the coprimality triangle are defined as

$T(n,i)\equiv [(n,i)=1],\ 1\leq i\leq n-1,\,$

where $\scriptstyle (n,i)\,$ is the greatest common divisor and $\scriptstyle [\cdot ]\,$ is the Iverson bracket.

Properties

Symmetry

The coprimality triangle is symmetric with respect to a central vertical line, i.e.

$T(n,i)=T(n,n-i)\,$

since

$(n,i)=(n,n-i).\,$

Note, among others, the patterns highlighted in blue, purple and green (symmetric about multiples of 6) of the noncoprime numbers.

Connectivity of noncoprimes?

It seems that if we form a graph where each node corresponds to a triangle cell with value 1 and each edge links nodes corresponding to adjacent cells with value 1 that the resulting graph is connected, i.e. has the connectivity property. Is that true and if so is there a proof?

Row sums

The row sums give Euler's totient function of 

n

(count of numbers less than 

n

and coprime to 

n

), i.e.

$\sum _{i=1}^{n-1}T(n,i)=\varphi (n).\,$

Sequences

The rows give an infinite sequence of finite sequences

 

{{1}, {1, 1}, {1, 0, 1}, {1, 1, 1, 1}, {1, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1}, {1, 0, 1, 0, 1, 0, 1}, {1, 1, 0, 1, 1, 0, 1, 1}, {1, 0, 1, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}, ...}

where the row sums give the sequence for Euler's totient function (A000010 

(n), n   ≥   2

)

 

{1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, ...}

whose partial sums plus one give the summatory function of Euler's totient function (A002088 

(n), n   ≥   2

)

 

{2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, }

and the concatenation of the rows gives the sequence (A054431)

 

{1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, ...}

whose partial sums plus one give (A?????? 

(n), n   ≥   2

)

 

{2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 11, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 30, 30, 30, 31, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 43, ...}

Relative prime triangle, $\scriptstyle T(n,i)\,\equiv \,i\cdot [(n,i)\,=\,1],\ 1\,\leq \,i\,\leq \,n,\,$

1; 1, 0; 1, 2, 0; 1, 0, 3, 0; 1, 2, 3, 4, 0; 1, 0, 0, 0, 5, 0; 1, 2, 3, 4, 5, 6, 0; ...

read by rows (A127368)

 

{1, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 2, 3, 4, 0, 1, 0, 0, 0, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 0, 3, 0, 5, 0, 7, 0, 1, 2, 0, 4, 5, 0, 7, 8, 0, 1, 0, 3, 0, 0, 0, 7, 9, 0 , ...}

See also

• Euler's totient function
• Strong coprimality triangle
• Weak coprimality triangle
• Divisibility triangle