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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.

%I A375358 #12 Sep 20 2024 06:41:48
%S A375358 2,2,14,26,46,74,106,134,194,206,266,286,346,374,382,442,454,506,550,
%T A375358 614,686,818,854,914,1034,1118,1186,1226,1274,1294,1606,1630,1618,
%U A375358 1702,1754,2018,2042,2078,2102,2174,2290,2434,2546,2534,2582,2626,2846,2890,2950
%N 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.
%C A375358 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|.
%e A375358 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.
%o A375358 (Python)
%o A375358 from itertools import islice
%o A375358 from sympy import isprime, nextprime, primerange
%o A375358 def agen(): # generator of terms
%o A375358     p, q = 2, 3
%o A375358     while True:
%o A375358         if q - p == 2:
%o A375358             s = p + q
%o A375358             yield max(m-k for k in primerange(2, s//2+1) if isprime(m:=s-k))
%o A375358         p, q = q, nextprime(q)
%o A375358 print(list(islice(agen(), 80))) # _Michael S. Branicky_, Aug 13 2024
%Y A375358 Cf. A001359, A020482, A014574.
%K A375358 nonn
%O A375358 1,1
%A A375358 _Gonzalo Martínez_, Aug 13 2024