This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348168 #24 Jul 08 2025 15:46:31
%S A348168 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,
%T A348168 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,
%U A348168 2,2,1,2,2,1,4,1,1,3,2,2,3,1,2,1,2,1,2
%N A348168 Segment the list of prime numbers into sublists L_1, L_2, ... with L_1 = {2} and L_n = {p_1, p_2, ..., p_a(n)}, where a(n) is the largest m such that for 0 < i < m, p_1 - prevprime(p_1) > p_2 - p_1 >= p_{i+1} - p_i.
%C A348168 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.
%C A348168 Conjecture 1: lim_{n->oo} N_i/n = k_i, where N_i is the number of the first n sublists consisting of i primes and k_i is a constant, with k_2 > k_1 > k_3 > k_4 > ... .
%C A348168 Conjecture 2: lim_{n->oo} (Sum_{i=1..n} a(i))/n = Sum_{i=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 the links).
%C A348168 From _Ya-Ping Lu_, Apr 15 2024: (Start)
%C A348168 The distribution of sublists with 1, 2, 3, 4 and 5 primes and the number of primes in the first n sublists are given in the table below. k_i's as defined in Conjecture 1 are: k1 = 0.281, k2 = 0.431, k3 = 0.127, k4 = 0.058, and k5 = 0.031, approximately. Sublists with length <= 5 account for about 93% of the terms and 70% of the primes, as n approaches infinity.
%C A348168 n N_1 N_2 N_3 N_4 N_5 # of primes
%C A348168 ---------- --------- --------- --------- -------- -------- -----------
%C A348168 1 1 0 0 0 0 1
%C A348168 10 6 3 0 1 0 16
%C A348168 100 33 44 5 9 3 232
%C A348168 1000 277 431 120 72 36 2617
%C A348168 10000 2821 4225 1243 642 331 27214
%C A348168 100000 28072 42929 12427 6059 3159 276081
%C A348168 1000000 279751 430299 126008 59729 32043 2747392
%C A348168 10000000 2804959 4303512 1264532 592726 317127 27426366
%C A348168 100000000 28070302 43078975 12686566 5869443 3143266 273972452
%C A348168 1000000000 280903920 431182582 127100032 58293618 31258182 2737643048
%C A348168 (End)
%H A348168 Ya-Ping Lu, <a href="/A348168/a348168.pdf">Partial average of A348168</a>
%H A348168 <a href="/index/Pri#gaps">Index entries for sequences related to gaps between primes</a>
%e A348168 See also the table of the sublists in the examples for A362017.
%e A348168 a(1) = 1 because L_1 = {2} by definition.
%e A348168 In the following examples we use p_0 to denote prevprime(p_1).
%e A348168 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}.
%e A348168 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}.
%e A348168 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.
%o A348168 (Python)
%o A348168 from sympy import nextprime
%o A348168 L = [2]
%o A348168 for n in range(1, 100):
%o A348168 print(len(L), end =', ')
%o A348168 p0 = L[-1]; p1 = nextprime(p0); M = [p1]; g0 = p1 - p0; p = nextprime(p1); g1 = p - p1
%o A348168 while g1 < g0 and p - p1 <= g1: M.append(p); p1 = p; p = nextprime(p)
%o A348168 L = M
%Y A348168 Cf. A362017 (first in each sublist), A087641, A226657, A001359, A023200.
%K A348168 nonn,easy
%O A348168 1,5
%A A348168 _Ya-Ping Lu_, Oct 03 2021
%E A348168 Edited by _Peter Munn_, Jul 08 2025