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

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

Offset: 0

Author

Antti Karttunen, Jan 11 2014

Keywords

• nonn

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 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.

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 non-oriented trees when this bijection is applied to them, as trees encoded by Matula-Goebel numbers (cf. A061773).

Links

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

Crossrefs

Inverse: A235486. Cf. also A235201, A235487, A235493/A235494, A234743/A234744, A000081, A061773.

Programs

Scheme

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

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.

Extensions

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