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 A336321 #41 Feb 12 2021 11:39:06
%S A336321 1,2,3,4,7,5,19,6,9,11,53,10,131,23,13,8,311,15,719,22,29,59,1619,14,
%T A336321 49,137,21,46,3671,17,8161,12,61,313,37,25,17863,727,139,26,38873,31,
%U A336321 84017,118,39,1621,180503,20,361,77,317,274,386093,33,71,58,733,3673,821641,34,1742537,8167,87,18,151,67,3681131,626,1627,41,7754077,35,16290047
%N A336321 a(n) = A122111(A225546(n)).
%C A336321 A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).
%C A336321 In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.
%H A336321 Michel Marcus, <a href="/A336321/b336321.txt">Table of n, a(n) for n = 1..148</a>
%H A336321 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A336321 a(n) = A122111(A225546(n)).
%F A336321 Alternative definition: (Start)
%F A336321 Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
%F A336321 a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
%F A336321 (End)
%F A336321 a(A000040(m)) = A033844(m-1).
%F A336321 a(A001146(m)) = 2^(m+1).
%F A336321 a(2^n) = A057335(n).
%F A336321 a(n^2) = A253560(a(n)).
%F A336321 For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
%F A336321 More generally, a(A334747(n)) = b(a(n)).
%F A336321 a(A003961(n)) = A297002(a(n)).
%F A336321 a(A334866(m)) = A253563(m).
%e A336321 From _Peter Munn_, Jan 04 2021: (Start)
%e A336321 In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
%e A336321 First, a table showing mapping of the powers of 2:
%e A336321 n a^-1(2^n) = 2^n = a(2^n) =
%e A336321 A001146(n-1) A000079(n) A057335(n)
%e A336321 0 (1) 1 1
%e A336321 1 2 2 2
%e A336321 2 4 4 4
%e A336321 3 16 8 6
%e A336321 4 256 16 8
%e A336321 5 65536 32 12
%e A336321 6 4294967296 64 18
%e A336321 ...
%e A336321 Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
%e A336321 n a^-1(A019565(n)) A019565(n) a(A019565(n)) a^2(A019565(n))
%e A336321 Cf. {A337533} Cf. {A005117} = prime(n) = A033844(n-1)
%e A336321 0 1 1 (1) (1)
%e A336321 1 2 2 2 2
%e A336321 2 3 3 3 3
%e A336321 3 8 6 5 7
%e A336321 4 6 5 7 19
%e A336321 5 12 10 11 53
%e A336321 6 18 15 13 131
%e A336321 7 128 30 17 311
%e A336321 8 5 7 19 719
%e A336321 9 24 14 23 1619
%e A336321 ...
%e A336321 As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
%e A336321 Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
%e A336321 (End)
%Y A336321 A122111 composed with A225546.
%Y A336321 Cf. A336322 (inverse permutation).
%Y A336321 Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
%Y A336321 Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
%Y A336321 Cf. A057335.
%Y A336321 A mapping between the binary tree sequences A334866 and A253563.
%Y A336321 Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).
%K A336321 nonn
%O A336321 1,2
%A A336321 _Antti Karttunen_ and _Peter Munn_, Jul 17 2020