A375358 a(n) is the greatest difference between m and k, with m, k both prime such that k + m = p + q, where (p, q) is the n-th twin prime pair.

2, 2, 14, 26, 46, 74, 106, 134, 194, 206, 266, 286, 346, 374, 382, 442, 454, 506, 550, 614, 686, 818, 854, 914, 1034, 1118, 1186, 1226, 1274, 1294, 1606, 1630, 1618, 1702, 1754, 2018, 2042, 2078, 2102, 2174, 2290, 2434, 2546, 2534, 2582, 2626, 2846, 2890, 2950

Offset: 1

Author

Gonzalo Martínez, Aug 13 2024

Keywords

• nonn

Comments

If p and q are twin primes and x is their average, then among all pairs of primes (k, m) such that |x - k| = |x - m|, it is observed that p and q are at the smallest distance from x, which is 1. Our interest lies in finding the pair (m, k) such that the distance to x is maximum and then determining |k - m|.

Examples

			Since the 3rd pair of twin primes is (11, 13), whose sum is 24, and the other pairs of primes that sum to 24 are (5, 19) and (7, 17), the greatest difference is 19 - 5 = 14. Therefore, a(3) = 14.
		

Crossrefs

Cf. A001359, A020482, A014574.

Programs

Python

from itertools import islice
from sympy import isprime, nextprime, primerange
def agen(): # generator of terms
    p, q = 2, 3
    while True:
        if q - p == 2:
            s = p + q
            yield max(m-k for k in primerange(2, s//2+1) if isprime(m:=s-k))
        p, q = q, nextprime(q)
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 13 2024