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 A234743 #27 Mar 13 2018 04:16:59
%S A234743 0,1,3,2,9,19,6,61,27,4,57,17,18,433,183,38,81,101,12,7,171,122,51,
%T A234743 173,54,361,1299,8,549,43,114,31,243,34,303,1159,36,1811,21,866,513,
%U A234743 733,366,157,153,76,519,613,162,3721,1083,202,3897,1193,24,323,1647,14,129,59,342,5
%N A234743 Multiplicative permutation of integers: a(n) = A235199(A234840(n)).
%C A234743 Consider two self-inverse and multiplicative permutations, b and c defined as follows:
%C A234743 b(0)=0, b(1)=1, b(2)=3, b(3)=2, b(p_i) = p_{b(i+1)-1} for primes with index i > 2, and b(u*v) = b(u)*b(v) for u, v > 0.
%C A234743 c(n)=n if n < 4, c(5)=7 and c(7)=5, c(p_i) = p_{c(i)} for primes with index i > 4, and c(u*v) = c(u)*c(v) for u, v > 0.
%C A234743 This permutation is defined as their composition: a(n) = c(b(n)) = A235199(A234840(n)).
%C A234743 It is also multiplicative: a(u*v) = c(b(u*v)) = c(b(u)*b(v)) = c(b(u))*c(b(v)) = a(u)*a(v). For primes p_i with index i, a(p_i) = c(b(p_i)) = c(p_{b(i+1)-1}) = p_{c(b(i+1)-1)} = A000040(A235047(i)), except for cases i=8 and i=18, use 7 and 5, instead of 5 and 7.
%C A234743 Because 22 = 2*11, and 2 is in a two-cycle and 11 is in a three-cycle, 22 is in a cycle whose length is lcm(2,3) = 6: a(22)=51 (= a(2)*a(11) = 3*17), a(51)=202, a(202)=33, a(33)=34, a(34)=303, a(303)=22.
%C A234743 Among primes, there are at least fixed points (31), two-cycles (2 <-> 3), (37 <-> 1811), three-cycles: (11, 17, 101), (29, 43, 157), four-cycles: (5, 19, 7, 61), (41, 733, 359, 1091), eight-cycles: (47, 613, 2593, 1163, 1733, 409, 73, 131).
%C A234743 How long is the cycle beginning from 13, a(13)=433, a(433)=20693, a(20693)=? or from 23? (23, 173, 24043, ...)
%C A234743 Question: Are there any infinite cycles? If there are, what is the ratio of terms (primes) in finite cycles vs. infinite cycles?
%H A234743 Antti Karttunen, <a href="/A234743/b234743.txt">Table of n, a(n) for n = 0..462</a>
%H A234743 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A234743 a(n) = A235199(A234840(n)).
%F A234743 A000035(a(n)) = A000035(A234840(n)) = A000035(A064614(n)) = A011655(n) = for all n.
%Y A234743 Inverse: A234744. Similarly composed multiplicative permutations, but with more tractable cycle structures: A235485/A235486, A235493/A235494.
%Y A234743 Cf. A234840, A235199, A235200, A235201, A235047.
%K A234743 nonn,mult,nice
%O A234743 0,3
%A A234743 _Antti Karttunen_, Jan 04 2014