The sequence is:
$$1,3,7,13,21,31,...$$
$$ OR $$
$$(1),(1+2),(1+2+4),(1+2+4+6), (1+2+4+6+8), (1+2+4+6+8+...),...$$
$$OR$$
$$X_i =i^2+i+1$$
$$OR$$
$$X_i = 1+2T_i$$
($T$ for triangular numbers) $$\begin{matrix} index&Sequence&factorization \\0&1& \\1&3& \\2&7& \\3&13& \\4&21&(3⋅7) ** \\5&31& \\6&43& \\7&57&(3⋅19) \\8&73& \\9&91&(7⋅13) ** \\10&111&(3⋅37) \\11&133& (7⋅19) \\12&157& \\13&183& (3⋅61) \\14&211& \\15&241& \\16&273& (13⋅21) ** \\17&307& \\18&343& (7^3) \\19&381& (3⋅127) \\20&421& \\21&463& \\22&507& (3 ⋅13^2) \\23&553& (7 ⋅79) \\24&601& \\25&651& (21⋅31) ** \\26&703& (19⋅37) \\.&.&. \\.&.&. \\.&.&. \end{matrix}$$
Let's say $X_i$ is a number in the sequence ($i$ for index) and its prime factors are $x_a$, then any $x_a$ will appear as a prime factor in $X_{i+x_a}$.
Once "introduced" the prime factors $x_a$ repeat themselves $x_a$ times, but they do get "introduced" more than once.
There is also the pattern that if $i$ is a square number such as $X_4 = X_1⋅X_2$, $X_9 = X_2⋅X_3$, $X_{16} = X_3⋅X_4$
Since every factor is "introduced" more than one, I was wondering, the following:
The subset in question is: $1,3,7,13,31,43,73,157,211,241,307,421,463,601,...$
These are the numbers that do not have factors that were "introduced" earlier.
Are these number always prime numbers (except $1$)? and will they continue for ever?