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A235485 Permutation of natural numbers: a(n) = A235201(A235487(n)).

%I A235485 #28 Feb 12 2018 17:41:45
%S A235485 0,1,2,4,3,7,8,6,5,16,14,17,12,19,9,28,10,13,32,11,21,24,34,53,20,49,
%T A235485 38,64,18,43,56,59,15,68,26,42,48,37,22,76,35,67,36,23,51,112,106,107,
%U A235485 40,27,98,52,57,29,128,119,30,44,86,41,84,131,118,96,25,133,136,31,39,212,63,73,80
%N A235485 Permutation of natural numbers: a(n) = A235201(A235487(n)).
%C A235485 The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=2 and n=4.
%C A235485 This permutation has only finite cycles: numbers 0, 1, 2, 3, ... are in the cycles of size: 1, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 2, 4, 4, 4, 4, 5, 5, 1, 4, 4, 5, 4, 5, 4, 4, 2, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 7, 4, 4, 7, 4, 4, 4, 4, 4, 4, 7, 3, 7, 3, 1, ...
%C A235485 The first number with cycle size 1 (i.e., fixed point) is 0, the first in a 2-cycle is 3 (as a(3) = 4, a(4) = 3), the first in 3-cycle is 20, the first in 4-cycle is 5, the first in 5-cycle is 35, in 6-cycle 213, in 7-cycle 60, in 8-cycle and 9-cycle (no terms among 0..10080), the first in 10-cycle: 447, the first in 12-cycle: 220, in 14-cycle: 843, in 15-cycle: 2485, in 20-cycle: 385.
%C A235485 Please compare to the cycle structure of A235493/A235494.
%C A235485 Also of interest is the number of separate cycles (orbits) and fixed points among each A000081(n) rooted non-oriented trees when this bijection is applied to them, as trees encoded by Matula-Goebel numbers (cf. A061773).
%H A235485 Antti Karttunen, <a href="/A235485/b235485.txt">Table of n, a(n) for n = 0..10080</a>
%H A235485 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%H A235485 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A235485 a(n) = A235201(A235487(n)).
%F A235485 As a recurrence:
%F A235485   a(0)=0, a(1)=1, a(2)=2,
%F A235485   a(3*n) = 4*a(n),
%F A235485   a(8*n) = 5*a(n),
%F A235485   a(4*n) = 3*a(n) [when n is odd],
%F A235485   a(14*n) = 9*a(n),
%F A235485   a(49*n) = 27*a(n),
%F A235485   a(7*n) = 6*a(n) [when n is odd and not divisible by 7],
%F A235485   a(p_i) = p_{a(i)} for primes whose index i is neither 2 nor 4 [primes other than 3 or 7],
%F A235485 and
%F A235485   a(u * v) = a(u) * a(v) for other composite cases.
%o A235485 (Scheme) (define (A235485 n) (A235201 (A235487 n)))
%o A235485 ;; Alternative implementation based on the given direct recurrence. Needs _Antti Karttunen_'s IntSeq-library:
%o A235485 (definec (A235485 n) (cond ((< n 3) n) ((zero? (modulo n 3)) (* 4 (A235485 (/ n 3)))) ((zero? (modulo n 8)) (* 5 (A235485 (/ n 8)))) ((zero? (modulo n 4)) (* 3 (A235485 (/ n 4)))) ((zero? (modulo n 14)) (* 9 (A235485 (/ n 14)))) ((zero? (modulo n 49)) (* 27 (A235485 (/ n 49)))) ((zero? (modulo n 7)) (* 6 (A235485 (/ n 7)))) ((= 1 (A010051 n)) (A000040 (A235485 (A000720 n)))) (else (reduce * 1 (map A235485 (ifactor n))))))
%Y A235485 Inverse: A235486. Cf. also A235201, A235487, A235493/A235494, A234743/A234744, A000081, A061773.
%K A235485 nonn
%O A235485 0,3
%A A235485 _Antti Karttunen_, Jan 11 2014
%E A235485 Name and incorrect claim about multiplicativity corrected by _Antti Karttunen_, Feb 12 2018