Table 8 Triangular and twisted triangular numbers

m, n : any positive integer

m, n, $\frac{m+n}{2}$ : odd prime numbers

$T_{m+n}=T_m+T_n+(mn)$

$T_{m+n}^{\ast }=T_m^{\ast }+T_n^{\ast }+mn+\frac{m+n}{2}$

[[13], (2)]

n : any positive integer

n : odd prime number

$T_{2n}-2T_n=n^2$

$T_{2^{m}n}^{\ast }-4^m{T}_n^{\ast }=0$

[[14], (1.13)]

n : any positive integer

2n − 1, n − 1 : odd prime numbers

$T_{2n-1}-2T_{n-1}=n^2$

$T_{2n-1}^{\ast }-2T_{n-1}^{\ast }=n^2$

[[14], (1.14)]

n : any positive integer

n, 5n + 2 : odd prime numbers

$25T_n+3=T_{5n+2}$

$25T_n^{\ast }+3=T_{5n+2}^{\ast }$

[[14], (4.2)]