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 A381169 #11 Mar 02 2025 23:54:20
%S A381169 1,1,1,1,1,1,2,2,2,3,2,2,1,1,2,1,6,3,2,2,2,1,1,5,2,2,2,3,1,2,2,2,2,3,
%T A381169 2,2,1,2,4,2,2,2,2,5,2,2,1,1,1,3,2,2,1,3,3,2,1,4,2,3,2,2,1,2,2,3,3,1,
%U A381169 3,2,1,2,1,1,2,3,3,1,1,2,2,3,2,2,1,5,2
%N A381169 List of twin prime averages (A014574) is partitioned by including as many elements as possible in the n-th partition, L_n, such that any gap in L_n is smaller than the gap between L_n and L_(n-1) but not bigger than the first gap in L_n. a(n) is the number of elements in L_n.
%C A381169 The partition method used here is the same as that in A348168.
%C A381169 Conjecture 1: lim_{n->oo} N_i/n = k_i, where N_i is the number of partitions with i elements and k_i is a constant, with k_2 > k_1 > k_3 > k_4 > .... The values of k_i are the same as those in A348168.
%C A381169 Conjecture 2: lim_{n->oo} Sum_{1..n} a(n)/n = lim_{i->oo} Sum_{1..i} i*k_i = e, or the average partition length approaches 2.71828... as n tends to infinity.
%C A381169 Numbers of twin prime pairs (N) and partitions with 1 through 6 twin prime pairs for n up to 10000000 are given in the table below.
%C A381169 n N N_1 N_2 N_3 N_4 N_5 N_6
%C A381169 -------- -------- ------- ------- ------- ------ ------ ------
%C A381169 1 1 1 0 0 0 0 0
%C A381169 10 15 6 3 1 0 0 0
%C A381169 100 209 30 45 16 5 3 1
%C A381169 1000 2536 286 416 145 64 29 19
%C A381169 10000 26474 2851 4331 1271 544 311 190
%C A381169 100000 271338 28034 43375 12923 5731 3002 1870
%C A381169 1000000 2725126 281837 434234 128190 56563 30074 18171
%C A381169 10000000 27120107 2815831 4352926 1276953 563128 302256 181612
%e A381169 Twin prime pair averages in the first 10 partitions are: [4], [6], [12], [18], [30], [42], [60, 72], [102, 108], [138, 150], and [180, 192, 198]. Thus, a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1, a(7) = a(8) = a(9) = 2, and a(10) = 3.
%o A381169 (Python)
%o A381169 from sympy import isprime, nextprime; L = [4]
%o A381169 def nexttwin(x):
%o A381169 p1 = nextprime(x); t1 = p1 + 2
%o A381169 while isprime(t1) == 0: p1 = nextprime(t1); t1 = p1 + 2
%o A381169 return p1+1
%o A381169 for _ in range(2, 89):
%o A381169 print(len(L), end = ', ')
%o A381169 t0 = L[-1]; t1 = nexttwin(t0); g0 = t1 - t0; M = [t1]; t = nexttwin(t1); g1 = t - t1
%o A381169 while g1 < g0 and t - t1 <= g1: M.append(t); t1 = t; t = nexttwin(t)
%o A381169 L = M
%Y A381169 Cf. A001097, A014574, A348168.
%K A381169 nonn
%O A381169 1,7
%A A381169 _Ya-Ping Lu_, Feb 15 2025