Search for a "greater no. of primes" sequence

Question

Is there any particular arithmetic progression of natural numbers in which the number of primes is greater than the number of composites till any term of the progression? (Either proven or conjectured). If I list the terms- a, a+d, a+2d, ......, m (where m = a + nd for some n). Then no matter which term I stop at, the number of primes encountered till then should be greater than the number of composites

Answer 1

No. It is known that $\pi(n)$, the number of primes not exceeding $n$, is asymptotically $\approx n/\ln n$. If an arithmetic progression with step $d$ would have a majority of primes, then we would have $$\liminf_{n\to\infty}\pi(n)/n\ge 1/(2d).$$

But $\pi/n\approx1/\ln n\to0$ as $n\to\infty$ so this is impossible.

Answer 2

Infinitely many. Consider $3$ - term $AP$. There is assertion that there are infinitely many twin primes. The last or the first term will always be divisible by $3$.

Example:

$29, 31, 33$

$27, 29, 31$