This site is supported by donations to The OEIS Foundation.

# Divisibility triangle

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

## Nontrivial divisors triangle

 ${\displaystyle n\,}$ ${\displaystyle s_{0}(n)-1=\,}$ ${\displaystyle \sigma _{0}(n)-2=\,}$ ${\displaystyle \sum _{i=2}^{{\big \lceil }{\frac {n}{2}}{\big \rceil }}T_{d}(n,i)\,}$ 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 ${\displaystyle \scriptstyle i\,}$ = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

## Nondivisors triangle

 ${\displaystyle n\,}$ ${\displaystyle {\overline {\sigma _{0}}}(n)=\,}$ ${\displaystyle n-\sigma _{0}(n)=\,}$ ${\displaystyle \sum _{i=1}^{n-1}T_{n}(n,i)\,}$ 2 0 ${\displaystyle \scriptstyle n\,}$ - 2 = 0 3 0 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 1 4 0 0 1 ${\displaystyle \scriptstyle n\,}$ - 3 = 1 5 0 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 3 6 0 0 0 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 2 7 0 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 5 8 0 0 1 0 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 4 9 0 1 0 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 3 = 6 10 0 0 1 1 0 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 6 11 0 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 9 12 0 0 0 0 1 0 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 6 = 6 13 0 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 11 14 0 0 1 1 1 1 0 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 10 15 0 1 0 1 0 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 11 16 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 5 = 11 17 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 15 18 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 6 = 12 19 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 17 20 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 6 = 14 21 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 17 22 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 6 = 26 ${\displaystyle \scriptstyle i\,}$ = 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

 ${\displaystyle n\,}$ ${\displaystyle {\overline {\sigma _{0}}}(n)=\,}$ ${\displaystyle n-\sigma _{0}(n)=\,}$ ${\displaystyle \sum _{i=1}^{n-1}T_{n}(n,i)\,}$ 2 0 ${\displaystyle \scriptstyle n\,}$ - 2 = 0 3 0 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 1 4 0 0 1 ${\displaystyle \scriptstyle n\,}$ - 3 = 1 5 0 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 3 6 0 0 0 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 2 7 0 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 5 8 0 0 1 0 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 4 9 0 1 0 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 3 = 6 10 0 0 1 1 0 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 6 11 0 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 9 12 0 0 0 0 1 0 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 6 = 6 13 0 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 11 14 0 0 1 1 1 1 0 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 10 15 0 1 0 1 0 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 11 16 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 5 = 11 17 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 15 18 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 6 = 12 19 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 2 = 17 20 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 6 = 14 21 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 4 = 17 22 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 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 ${\displaystyle \scriptstyle n\,}$ - 6 = 26 ${\displaystyle i\,}$ = 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

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

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

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

### Nondivisors triangle

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

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

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

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

## 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 ${\displaystyle \scriptstyle n\,}$) of ${\displaystyle \scriptstyle n,\ n\geq 3,\,}$ i.e.

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

The row sums plus two of the nontrivial divisors triangle give the number of divisors of ${\displaystyle \scriptstyle n\,}$, 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, ...}
${\displaystyle G_{\{\sigma _{0}(n)\}}(x)\equiv \sum _{n=1}^{\infty }\sigma _{0}(n)\,x^{n}=\sum _{n=0}^{\infty }{\frac {x^{n}}{(1-x^{n})}}.\,}$

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 ${\displaystyle \scriptstyle n\,}$) of ${\displaystyle \scriptstyle n,\ n\geq 2,\,}$ i.e.

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

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