Table 11 ${\mathit{\sigma }}_{\mathbf{1}}^{\mathbf{\ast }}\mathbf{(}\mathit{n}\mathbf{)}$ and ${\mathit{\sigma }}_{\mathbf{1}}\mathbf{(}\mathit{n}\mathbf{)}$

n<200

3, 6, 7, 10, 22, 31, 58, 82, 106, 140, 154, 160, 166, 180

14

n<4.25 × 10⁹

$\mathrm{♯}\{n|{\sigma }_1^{\ast }(n)={\sigma }_1^{\ast }(n+1)\}=1\text{,}870$

$\mathrm{♯}\{n|{\sigma }_1(n)={\sigma }_1(n+1)\}=832$

n

${\sigma }_1^{\ast }(n)={\sigma }_1^{\ast }(n+1)={\sigma }_1^{\ast }(n+2)$

${\sigma }_1(n)={\sigma }_1(n+1)={\sigma }_1(n+2)$

n<200

6

no

n<4.25 × 10⁹

6

no

n

${\sigma }_1^{\ast }(n+2)={\sigma }_1^{\ast }(n)+2$

${\sigma }_1(n+2)={\sigma }_1(n)+2$

twin prime

all

all

m, n

$m{\sigma }_1^{\ast }(m)=n{\sigma }_1^{\ast }(n)$, m ≠ n

$m{\sigma }_1(m)=n{\sigma }_1(n)$

1 ≤ m,n ≤ 7 × 10⁴

no

m = 12q, n = 14q (q,42)=1