Divisibility triangle

The divisibility triangle shows the divisors of a number by their positions in a triangle.

Nontrivial divisors triangle

The equilateral version of the divisibility triangle
(Nontrivial divisors triangle)
$n\,$																														$s_{0}(n)-1=\,$ $\sigma _{0}(n)-2=\,$ $\sum _{i=2}^{{\big \lceil }{\frac {n}{2}}{\big \rceil }}T_{d}(n,i)\,$
3															0															2 - 2 = 0
4															1															3 - 2 = 1
5														0		0														2 - 2 = 0
6														1		1														4 - 2 = 2
7													0		0		0													2 - 2 = 0
8													1		0		1													4 - 2 = 2
9												0		1		0		0												3 - 2 = 1
10												1		0		0		1												4 - 2 = 2
11											0		0		0		0		0											2 - 2 = 0
12											1		1		1		0		1											6 - 2 = 4
13										0		0		0		0		0		0										2 - 2 = 0
14										1		0		0		0		0		1										4 - 2 = 2
15									0		1		0		1		0		0		0									4 - 2 = 2
16									1		0		1		0		0		0		1									5 - 2 = 3
17								0		0		0		0		0		0		0		0								2 - 2 = 0
18								1		1		0		0		1		0		0		1								6 - 2 = 4
19							0		0		0		0		0		0		0		0		0							2 - 2 = 0
20							1		0		1		1		0		0		0		0		1							6 - 2 = 4
21						0		1		0		0		0		1		0		0		0		0						4 - 2 = 2
22						1		0		0		0		0		0		0		0		0		1						4 - 2 = 2
23					0		0		0		0		0		0		0		0		0		0		0					2 - 2 = 0
24					1		1		1		0		1		0		1		0		0		0		1					8 - 2 = 6
25				0		0		0		1		0		0		0		0		0		0		0		0				3 - 2 = 1
26				1		0		0		0		0		0		0		0		0		0		0		1				4 - 2 = 2
27			0		1		0		0		0		0		0		1		0		0		0		0		0			4 - 2 = 2
28			1		0		1		0		0		1		0		0		0		0		0		0		1			6 - 2 = 4
29		0		0		0		0		0		0		0		0		0		0		0		0		0		0		2 - 2 = 0
30		1		1		0		1		1		0		0		0		1		0		0		0		0		1		8 - 2 = 6
31	0		0		0		0		0		0		0		0		0		0		0		0		0		0		0	2 - 2 = 0
32	1		0		1		0		0		0		1		0		0		0		0		0		0		0		1	6 - 2 = 4
	$\scriptstyle i\,$ = 2		3		4		5		6		7		8		9		10		11		12		13		14		15		16

Nondivisors triangle

The equilateral version of the divisibility triangle
(Nondivisors triangle)
$n\,$																																																														${\overline {\sigma _{0}}}(n)=\,$ $n-\sigma _{0}(n)=\,$ $\sum _{i=1}^{n-1}T_{n}(n,i)\,$
2																															0																															$\scriptstyle n\,$ - 2 = 0
3																														0		1																														$\scriptstyle n\,$ - 2 = 1
4																													0		0		1																													$\scriptstyle n\,$ - 3 = 1
5																												0		1		1		1																												$\scriptstyle n\,$ - 2 = 3
6																											0		0		0		1		1																											$\scriptstyle n\,$ - 4 = 2
7																										0		1		1		1		1		1																										$\scriptstyle n\,$ - 2 = 5
8																									0		0		1		0		1		1		1																									$\scriptstyle n\,$ - 4 = 4
9																								0		1		0		1		1		1		1		1																								$\scriptstyle n\,$ - 3 = 6
10																							0		0		1		1		0		1		1		1		1																							$\scriptstyle n\,$ - 4 = 6
11																						0		1		1		1		1		1		1		1		1		1																						$\scriptstyle n\,$ - 2 = 9
12																					0		0		0		0		1		0		1		1		1		1		1																					$\scriptstyle n\,$ - 6 = 6
13																				0		1		1		1		1		1		1		1		1		1		1		1																				$\scriptstyle n\,$ - 2 = 11
14																			0		0		1		1		1		1		0		1		1		1		1		1		1																			$\scriptstyle n\,$ - 4 = 10
15																		0		1		0		1		0		1		1		1		1		1		1		1		1		1																		$\scriptstyle n\,$ - 4 = 11
16																	0		0		1		0		1		1		1		0		1		1		1		1		1		1		1																	$\scriptstyle n\,$ - 5 = 11
17																0		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1																$\scriptstyle n\,$ - 2 = 15
18															0		0		0		1		1		0		1		1		0		1		1		1		1		1		1		1		1															$\scriptstyle n\,$ - 6 = 12
19														0		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1														$\scriptstyle n\,$ - 2 = 17
20													0		0		1		0		0		1		1		1		1		0		1		1		1		1		1		1		1		1		1													$\scriptstyle n\,$ - 6 = 14
21												0		1		0		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1		1												$\scriptstyle n\,$ - 4 = 17
22											0		0		1		1		1		1		1		1		1		1		0		1		1		1		1		1		1		1		1		1		1											$\scriptstyle n\,$ - 4 = 18
23										0		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1										$\scriptstyle n\,$ - 2 = 21
24									0		0		0		0		1		0		1		0		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1									$\scriptstyle n\,$ - 8 = 16
25								0		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1								$\scriptstyle n\,$ - 3 = 22
26							0		0		1		1		1		1		1		1		1		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1							$\scriptstyle n\,$ - 4 = 22
27						0		1		0		1		1		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1						$\scriptstyle n\,$ - 4 = 23
28					0		0		1		0		1		1		0		1		1		1		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1		1					$\scriptstyle n\,$ - 6 = 22
29				0		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				$\scriptstyle n\,$ - 2 = 27
30			0		0		0		1		0		0		1		1		1		0		1		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1		1		1			$\scriptstyle n\,$ - 8 = 22
31		0		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		$\scriptstyle n\,$ - 2 = 29
32	0		0		1		0		1		1		1		0		1		1		1		1		1		1		1		0		1		1		1		1		1		1		1		1		1		1		1		1		1		1		1	$\scriptstyle n\,$ - 6 = 26
	$\scriptstyle i\,$ = 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

The rectangular version of the divisibility triangle
(Nondivisors triangle)
$n\,$																																${\overline {\sigma _{0}}}(n)=\,$ $n-\sigma _{0}(n)=\,$ $\sum _{i=1}^{n-1}T_{n}(n,i)\,$
2	0																															$\scriptstyle n\,$ - 2 = 0
3	0	1																														$\scriptstyle n\,$ - 2 = 1
4	0	0	1																													$\scriptstyle n\,$ - 3 = 1
5	0	1	1	1																												$\scriptstyle n\,$ - 2 = 3
6	0	0	0	1	1																											$\scriptstyle n\,$ - 4 = 2
7	0	1	1	1	1	1																										$\scriptstyle n\,$ - 2 = 5
8	0	0	1	0	1	1	1																									$\scriptstyle n\,$ - 4 = 4
9	0	1	0	1	1	1	1	1																								$\scriptstyle n\,$ - 3 = 6
10	0	0	1	1	0	1	1	1	1																							$\scriptstyle n\,$ - 4 = 6
11	0	1	1	1	1	1	1	1	1	1																						$\scriptstyle n\,$ - 2 = 9
12	0	0	0	0	1	0	1	1	1	1	1																					$\scriptstyle n\,$ - 6 = 6
13	0	1	1	1	1	1	1	1	1	1	1	1																				$\scriptstyle n\,$ - 2 = 11
14	0	0	1	1	1	1	0	1	1	1	1	1	1																			$\scriptstyle n\,$ - 4 = 10
15	0	1	0	1	0	1	1	1	1	1	1	1	1	1																		$\scriptstyle n\,$ - 4 = 11
16	0	0	1	0	1	1	1	0	1	1	1	1	1	1	1																	$\scriptstyle n\,$ - 5 = 11
17	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1																$\scriptstyle n\,$ - 2 = 15
18	0	0	0	1	1	0	1	1	0	1	1	1	1	1	1	1	1															$\scriptstyle n\,$ - 6 = 12
19	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1														$\scriptstyle n\,$ - 2 = 17
20	0	0	1	0	0	1	1	1	1	0	1	1	1	1	1	1	1	1	1													$\scriptstyle n\,$ - 6 = 14
21	0	1	0	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1												$\scriptstyle n\,$ - 4 = 17
22	0	0	1	1	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1											$\scriptstyle n\,$ - 4 = 18
23	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1										$\scriptstyle n\,$ - 2 = 21
24	0	0	0	0	1	0	1	0	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1									$\scriptstyle n\,$ - 8 = 16
25	0	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1								$\scriptstyle n\,$ - 3 = 22
26	0	0	1	1	1	1	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1							$\scriptstyle n\,$ - 4 = 22
27	0	1	0	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1						$\scriptstyle n\,$ - 4 = 23
28	0	0	1	0	1	1	0	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1					$\scriptstyle n\,$ - 6 = 22
29	0	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				$\scriptstyle n\,$ - 2 = 27
30	0	0	0	1	0	0	1	1	1	0	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1			$\scriptstyle n\,$ - 8 = 22
31	0	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		$\scriptstyle n\,$ - 2 = 29
32	0	0	1	0	1	1	1	0	1	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	$\scriptstyle n\,$ - 6 = 26
	$i\,$ = 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

Definitions

Nontrivial divisors triangle

The entries of the divisibility triangle (nontrivial divisors triangle) are defined as

T_{d}(n,i)\equiv [i\,|\,n]=[n~{\bmod {~}}i=0],\ 2\leq i\leq {\bigg \lceil }{\frac {n}{2}}{\bigg \rceil },\,

where $\scriptstyle n{\bmod {i}}\,$ is $\scriptstyle n\,$ modulo $\scriptstyle i\,$ and $\scriptstyle [\cdot ]\,$ is the Iverson bracket.

Thus the cells valued 1 correspond to nontrivial divisors (excluding 1 and $\scriptstyle n\,$ ) of $\scriptstyle n\,$ .

Nondivisors triangle

The entries of the divisibility triangle (nondivisors triangle) are defined as

T_{n}(n,i)\equiv [i\nmid n]=[n~{\bmod {~}}i\neq 0],\ 1\leq i\leq n-1,\,

where $\scriptstyle n{\bmod {i}}\,$ is $\scriptstyle n\,$ modulo $\scriptstyle i\,$ and $\scriptstyle [\cdot ]\,$ is the Iverson bracket.

Thus the cells valued 1 correspond to nondivisors of $\scriptstyle n\,$ .

Properties

Row sums

Row sums of nontrivial divisors triangle

The row sums of the nontrivial divisors triangle give the number of nontrivial divisors (excluding 1 and $\scriptstyle n\,$ ) of $\scriptstyle n,\ n\geq 3,\,$ i.e.

\sum _{i=2}^{{\big \lceil }{\frac {n}{2}}{\big \rceil }}T_{d}(n,i)=\sigma _{0}(n)-2.\,

The row sums plus two of the nontrivial divisors triangle give the number of divisors of $\scriptstyle n\,$ , producing the sequence (Cf. OEIS: A000005(n), n ≥ 3)

{2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, ...}

with generating function

G_{\{\sigma _{0}(n)\}}(x)\equiv \sum _{n=1}^{\infty }\sigma _{0}(n)\,x^{n}=\sum _{n=0}^{\infty }{\frac {x^{n}}{(1-x^{n})}}.\,

This is usually called THE Lambert series (see Knopp, Titchmarsh).

Row sums of nondivisors triangle

The row sums of the nondivisors triangle give the number of nondivisors (which are less than $\scriptstyle n\,$ ) of $\scriptstyle n,\ n\geq 2,\,$ i.e.

\sum _{i=1}^{n-1}T_{n}(n,i)={\overline {\sigma _{0}}}(n)=n-\sigma _{0}(n).\,

producing the sequence (Cf. OEIS: A049820(n), n ≥ 2)

{0, 1, 1, 3, 2, 5, 4, 6, 6, 9, 6, 11, 10, 11, 11, 15, 12, 17, 14, 17, 18, 21, 16, 22, 22, 23, 22, 27, 22, 29, 26, 29, 30, 31, 27, 35, 34, 35, 32, 39, 34, 41, 38, 39, 42, 45, 38, 46, 44, 47, 46, 51, 46, 51, 48, 53, 54, 57, ...}

with generating function

G_{\{{\overline {\sigma _{0}}}(n)\}}(x)\equiv \sum _{n=1}^{\infty }{\overline {\sigma _{0}}}(n)\,x^{n}=\sum _{n=1}^{\infty }{\frac {x^{2n+1}}{(1-x^{n})(1-x^{n+1})}}=\sum _{n=1}^{\infty }{\frac {x^{n}}{(1-x^{n})}}\,{\frac {x^{n+1}}{(1-x^{n+1})}}\,

References

K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
E. C. Titchmarsh, On a series of Lambert type, J. London Math. Soc., 13 (1938), 248-253.

Divisibility triangle

Contents

Nontrivial divisors triangle

Nondivisors triangle

Definitions

Nontrivial divisors triangle

Nondivisors triangle

Properties

Row sums

Row sums of nontrivial divisors triangle

Row sums of nondivisors triangle

See also

References

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