%I #4 Apr 23 2023 16:03:30

%S 2,3,5,7,11,17,23,29,37,53,59,67,79,89,97,127,137,149,157,163,173,179,

%T 191,197,211,223,239,251,293,307,331,347,353,359,367,397,409,419,431,

%U 439,449,457,479,521,541,557,587,631,673,683,691,701,719,787,809,821

%N a(n) is the leading prime in the n-th prime sublist defined in A348168.

%C If Conjecture 2 in A348168 is true, lim_{n->infinity} a(n)/prime(round((n-1)*e)+1) = 1, where e is Euler's number.

%C If a term p (>2) is from a single-prime sublist (A356271), then p is a weak prime (A051635) or a balanced prime (A006562). Otherwise, p is a strong prime (A051634).

%e According to the definition in A348168, prime numbers are divided into sublists, L_1, L_2, L_3,..., in which L_n = [p(n,1), p(n,2), ..., p(n,m(n))], where p(n,k) is the k-th prime and m(n) the number of primes in the n-th sublist L_n. Thus, a(n) = p(n,1). The first sublist L_1 = [2]. If p(n,1) <= (prevprime(p(n,1)) + nextprime(p(n,1)))/2, then L_n has only 1 prime, p(n,1). Otherwise, m(n) is the largest integer such that g(n,1) >= g(n,i), where g(n,i) = p(n,i+1) - p(n,i) and 2 <= i <= m(n).

%e The first 32 primes, for example, are divided into 16 prime sublists: [2], [3], [5], [7], [11,13], [17,19], [23], [29,31], [37,41,43,47], [53], [59,61], [67,71,73], [79,83], [89], [97,101,103,107,109,113], [127,131]. The leading primes in these sublists are: 2, 3, 5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127. Therefore, a(1) = 2, a(2) = 3, ..., and a(16) = 127.

%o (Python)

%o from sympy import nextprime; R = [2]; L = [2]

%o for n in range(2, 57):

%o p0 = L[-1]; p1 = nextprime(p0); M = [p1]; g0 = p1-p0; p = nextprime(p1); g1 = p-p1

%o while g1 < g0 and p-p1 <= g1: M.append(p); p1 = p; p = nextprime(p)

%o L = M; R.append(L[0])

%o print(*R, sep =', ')

%Y Cf. A006562, A051634, A051635, A348168, A356271.

%K nonn

%O 1,1

%A _Ya-Ping Lu_, Apr 04 2023