

A235485


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


10



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, 38, 64, 18, 43, 56, 59, 15, 68, 26, 42, 48, 37, 22, 76, 35, 67, 36, 23, 51, 112, 106, 107, 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
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OFFSET

0,3


COMMENTS

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=2 and n=4.
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, ...
The first number with cycle size 1 (i.e., fixed point) is 0, the first in a 2cycle is 3 (as a(3) = 4, a(4) = 3), the first in 3cycle is 20, the first in 4cycle is 5, the first in 5cycle is 35, in 6cycle 213, in 7cycle 60, in 8cycle and 9cycle (no terms among 0..10080), the first in 10cycle: 447, the first in 12cycle: 220, in 14cycle: 843, in 15cycle: 2485, in 20cycle: 385.
Please compare to the cycle structure of A235493/A235494.
Also of interest is the number of separate cycles (orbits) and fixed points among each A000081(n) rooted nonoriented trees when this bijection is applied to them, as trees encoded by MatulaGoebel numbers (cf. A061773).


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to MatulaGoebel numbers
Index entries for sequences that are permutations of the natural numbers


FORMULA

a(n) = A235201(A235487(n)).
As a recurrence:
a(0)=0, a(1)=1, a(2)=2,
a(3*n) = 4*a(n),
a(8*n) = 5*a(n),
a(4*n) = 3*a(n) [when n is odd],
a(14*n) = 9*a(n),
a(49*n) = 27*a(n),
a(7*n) = 6*a(n) [when n is odd and not divisible by 7],
a(p_i) = p_{a(i)} for primes whose index i is neither 2 nor 4 [primes other than 3 or 7],
and
a(u * v) = a(u) * a(v) for other composite cases.


PROG

(Scheme) (define (A235485 n) (A235201 (A235487 n)))
;; Alternative implementation based on the given direct recurrence. Needs Antti Karttunen's IntSeqlibrary:
(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))))))


CROSSREFS

Inverse: A235486. Cf. also A235201, A235487, A235493/A235494, A234743/A234744, A000081, A061773.
Sequence in context: A292958 A235493 A105081 * A026167 A127002 A027634
Adjacent sequences: A235482 A235483 A235484 * A235486 A235487 A235488


KEYWORD

nonn


AUTHOR

Antti Karttunen, Jan 11 2014


EXTENSIONS

Name and incorrect claim about multiplicativity corrected by Antti Karttunen, Feb 12 2018


STATUS

approved



