A348168 The list of all prime numbers is split into sublists with the 1st sublist L_1 = {2} and n-th sublist L_n = {p_1, p_2, ..., p_m}. a(n) is the largest m such that p_1 - p_0 > p_2 - p_1 = g, where p_0 = prevprime(p_1) and g is the maximum prime gap in L_n.

1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 2, 3, 2, 1, 6, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 8, 1, 4, 2, 2, 1, 1, 5, 2, 1, 2, 2, 2, 1, 4, 6, 2, 2, 5, 8, 7, 2, 1, 1, 2, 10, 2, 2, 2, 2, 1, 4, 4, 2, 1, 5, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 4, 1, 1, 3, 2, 2, 3, 1, 2, 1, 2, 1, 2

Offset

1,5

Comments

The gap between two consecutive primes in L_n is smaller than g_n-1 and g_n, where g_n is the gap between L_n and L_n+1. Sublists of length 2 are the most frequent ones and any pair of twin primes >= 11 stay in the same sublist.

Conjecture 1: lim_{n->oo} N_i/n = k_i, where N_i is the number of sublists consisting of i primes and k_i is a constant, with k_2 > k_1 > k_3 > k_4 > ....

Conjecture 2: lim_{n->oo} (Sum_{1..n} a(i))/n = Sum_{1..oo} i*k_i = e, meaning that, as n tends to infinity, the average length of sublists approaches 2.71828...(see the partial average - n plot in Links).

Links

Table of n, a(n) for n=1..87.

Ya-Ping Lu, Partial average of A348168

Example

a(1) = 1 because L_1 = {2} by definition.
a(2) = 1. For the 2nd sublist, p_1 - p_0 = 3 - 2 = 1. If the next prime, 5, is in L_2, then p_2 - p_1 = 2 > p_1 - p_0. Therefore, 5 does not belong to L_2 and L_2 = {3}.
a(5) = 2. For the 5th sublist, p_1 - p_0 = 11 - 7 = 4. p_2 = 13 is in L_5 because p_2 - p_1 = 2 < p_1 - p_0. However, the next prime, 17, is not in L_5 as 17 - 13 > p_2 - p_1. Thus, L_5 = {11, 13}.
a(15) = 6. L_15 = {97, 101, 103, 107, 109, 113}, because p_1 - p_0 = 97-89 > p_2 - p_1 = 101-97 = 4, which is the maximum prime gap in L_15. 127, the prime after 113, is not in L_15 as 127-113 = 14 > p_2 - p_1.

Prog

(Python)
from sympy import nextprime
L = [2]
for n in range(1, 100):
    print(len(L), end =', ')
    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

Crossrefs

Cf. A035789, A035790, A035791, A035792, A035793, A035794, A035795.

Cf. A087641, A226657, A229030, A001359, A006512, A023200, A046132, A023201, A046117.

Sequence in context: A327201 A021456 A125912 * A348178 A127677 A238009

Adjacent sequences: A348165 A348166 A348167 * A348169 A348170 A348171

Keyword

nonn

Author

Ya-Ping Lu, Oct 03 2021

Status

approved