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 A349406 #14 Jun 20 2025 17:40:41
%S A349406 19,3,5,3,2,2,7,3,3,2,3,5,3,2,3,3,2,2,2,3,2,2,3,2,5,2,5,2,2,2,3,7,2,2,
%T A349406 3,3,2,2,2,3,2,2,2,2,2,3,11,3,2,2,2,2,2,41,2,2,3,2,2,2,3,3,2,3,2,2,2,
%U A349406 5,3,5,3,3,5,2,2,2,2,2,2,2,2,3,5,3,2,2
%N A349406 a(n) = A349405(n)/A348779(n).
%C A349406 Ratio of progenitor and prime in A347113.
%C A349406 Let s = A347113, j = s(i)+1, and k = s(i+1). For prime k, j is a squarefree semiprime pq, p < q.
%C A349406 The first 3 primes in s have k = p, while all others observed for i <= 2^19 have k = q. This sequence thus lists the other prime factor r of j such that r*k = j.
%C A349406 The quasi-linear striations k < n are arranged according to this sequence (see color-coded log-log scatterplot). - _Michael De Vlieger_, Nov 17 2021
%H A349406 Michael De Vlieger, <a href="/A349406/b349406.txt">Table of n, a(n) for n = 1..10000</a>
%H A349406 Michael De Vlieger, <a href="/A349406/a349406.txt">Notes on ratio j/k in A347113</a>, listing numbers of certain ratios seen given 2^19 terms of A347113.
%H A349406 Michael De Vlieger, <a href="/A349406/a349406_1.txt">Extended table of n, a(n) for n = 1..23082</a> (all terms resulting from 2^19 terms of A347113)
%H A349406 Michael De Vlieger, <a href="/A349406/a349406.png">Log-log scatterplot of A347113(n), n=1..256</a> indicating primes in green, labeling them with the ratio j/k = a(n) = (A347113(n-1)+1)/A347113(n). Red dots represent records A347307 in A347113, and blue represent local minima A347756 in A347113.
%H A349406 Michael De Vlieger, <a href="/A349406/a349406_1.png">Log-log scatterplot of a(n)</a>, for n=1..2^16 showing primes k=A347113(A348784) listed in A348779, color coded according to the ratio j/k, which is always prime. Color function: red = 2, orange = 3, chartreuse = 5, small green = 7, large green = 11, large cyan = 13, large blue = 17, large indigo = 19, large purple = 29, large magenta = 41.
%e A349406 s(6)+1 = 95 -> s(7) = 5; a(1) = 95/5 = 19.
%e A349406 s(7)+1 = 6 -> s(8) = 2; a(2) = 6/2 = 3.
%e A349406 s(10)+1 = 15, -> s(11) = 3; a(3) = 15/3 = 5.
%e A349406 s(18)+1 = 39, -> s(19) = 13; a(4) = 39/13 = 3, etc.
%t A349406 c[_] = 0; j = m = 2; m = {1}~Join~Reap[Do[If[IntegerQ@ Log2[i], While[c[m] > 0, m++]]; Set[k, m]; While[Or[c[k] > 0, k == j, GCD[j, k] == 1], k++]; Sow[k]; Set[c[k], i]; j = k + 1, {i, 900}]][[-1, -1]]; Map[(#1 + 1)/#2 & @@ m[[# - 1 ;; #]] &, Position[m, _?PrimeQ][[All, 1]]]
%Y A349406 Cf. A006530, A020639, A347113, A347313, A348779, A349405.
%K A349406 nonn
%O A349406 1,1
%A A349406 _Michael De Vlieger_, Nov 16 2021