%I #41 Feb 12 2021 11:39:06
%S 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 49,137,21,46,3671,17,8161,12,61,313,37,25,17863,727,139,26,38873,31,
%U 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 a(n) = A122111(A225546(n)).
%C 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 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 Michel Marcus, <a href="/A336321/b336321.txt">Table of n, a(n) for n = 1..148</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = A122111(A225546(n)).
%F Alternative definition: (Start)
%F Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
%F a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
%F (End)
%F a(A000040(m)) = A033844(m-1).
%F a(A001146(m)) = 2^(m+1).
%F a(2^n) = A057335(n).
%F a(n^2) = A253560(a(n)).
%F For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
%F More generally, a(A334747(n)) = b(a(n)).
%F a(A003961(n)) = A297002(a(n)).
%F a(A334866(m)) = A253563(m).
%e From _Peter Munn_, Jan 04 2021: (Start)
%e In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
%e First, a table showing mapping of the powers of 2:
%e n a^-1(2^n) = 2^n = a(2^n) =
%e A001146(n-1) A000079(n) A057335(n)
%e 0 (1) 1 1
%e 1 2 2 2
%e 2 4 4 4
%e 3 16 8 6
%e 4 256 16 8
%e 5 65536 32 12
%e 6 4294967296 64 18
%e ...
%e Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
%e n a^-1(A019565(n)) A019565(n) a(A019565(n)) a^2(A019565(n))
%e Cf. {A337533} Cf. {A005117} = prime(n) = A033844(n-1)
%e 0 1 1 (1) (1)
%e 1 2 2 2 2
%e 2 3 3 3 3
%e 3 8 6 5 7
%e 4 6 5 7 19
%e 5 12 10 11 53
%e 6 18 15 13 131
%e 7 128 30 17 311
%e 8 5 7 19 719
%e 9 24 14 23 1619
%e ...
%e As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
%e Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
%e (End)
%Y A122111 composed with A225546.
%Y Cf. A336322 (inverse permutation).
%Y Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
%Y Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
%Y Cf. A057335.
%Y A mapping between the binary tree sequences A334866 and A253563.
%Y 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 nonn
%O 1,2
%A _Antti Karttunen_ and _Peter Munn_, Jul 17 2020