Starting at prime $23$, $$ 23 + 3 \cdot n \cdot(n+1) $$ is prime for $n=1$ to $21$. Is there a starting prime with more successes? Does this suggest that $23$ from $n=1$ to $1000$ would have the most successes? If set to n=1 to 10000, would it have the most successes? Does anyone know if extensive computer work has been done on these questions?
2026-03-27 18:05:46.1774634746
23 and arithmetic progression
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Let $f_m(p)$ be the number of primes of the form $p + 3n(n+1)$ for $0 \le n \le m$. Then $f_{1000}(23) = 442$, while $f_{1000}(103) = 472$, $f_{1000}(233) = 474$, and $f_{1000}(1663) = 479$.
I haven't tried $10000$, but again I would expect that there are some that do better than $23$.