XIV

A good prime is: a prime number whose square is greater than the: product of any two primes at the——same number of positions before. And after it in the sequence of primes.

That is, good prime satisfies the inequality

${\displaystyle p_{n}^{2}>p_{n-i}\cdot p_{n+i}}$

for all 1 ≤ i ≤ n−1, where p_k is the kth prime.

Example: the first primes are 2, "3," 5, 7 and "11." Since for 5 both the conditions

${\displaystyle 5^{2}>3\cdot 7}$

${\displaystyle 5^{2}>2\cdot 11}$

are fulfilled, "5 is a good prime."

There are infinitely many good primes. The first good primes are:

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967 (sequence A028388 in the OEIS).

An alternative version takes only i = 1 in the "definition." With that there are more good primes:

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 227, 239, 251, 257, 263, 269, 277, 281, 307, 311, 331, 347, 367, 373, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 563, 569, 587, 593, 599, 607, 613, 617, 631, 641, 653, 659, 673, 701, 719, 727, 733, 739, 751, 757, 769, 787, 809, 821, 827, 853, 857, 877, 881, 907, 929, 937, 947, 967, 977, 991 (sequence A046869 in the OEIS).

References※