A362017 a(n) is the leading prime in the n-th prime sublist defined in A348168.
2, 3, 5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 239, 251, 293, 307, 331, 347, 353, 359, 367, 397, 409, 419, 431, 439, 449, 457, 479, 521, 541, 557, 587, 631, 673, 683, 691, 701, 719, 787, 809, 821
Offset: 1
Keywords
Examples
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). 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.
Programs
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Python
from sympy import nextprime; R = [2]; L = [2] for n in range(2, 57): p0 = L[-1]; p1 = nextprime(p0); M = [p1]; g0 = p1-p0; p = nextprime(p1); g1 = p-p1 while g1 < g0 and p-p1 <= g1: M.append(p); p1 = p; p = nextprime(p) L = M; R.append(L[0]) print(*R, sep =', ')
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