A235487 -id:A235487 - cp's OEIS frontend

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

Antti Karttunen, Jan 11 2014

nonn

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).

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

(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))))))

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.

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

A235486 Permutation of natural numbers: a(n) = A235487(A235201(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 7, 5, 6, 14, 16, 19, 12, 17, 10, 32, 9, 11, 28, 13, 24, 20, 38, 43, 21, 64, 34, 49, 15, 53, 56, 67, 18, 76, 22, 40, 42, 37, 26, 68, 48, 59, 35, 29, 57, 112, 86, 131, 36, 25, 128, 44, 51, 23, 98, 152, 30, 52, 106, 31, 96, 107, 134, 70, 27, 136, 133, 41, 33, 172, 80, 89, 84

Offset: 0

Antti Karttunen, Jan 11 2014

nonn

See comments at A235485.

Inverse: A235485. Cf. also A235201, A235487, A235493/A235494.

(define (A235486 n) (A235487 (A235201 n)))

a(n) = A235487(A235201(n)).

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

A234840 Self-inverse and multiplicative permutation of integers: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 2, a(p_i) = p_{a(i+1)-1} for primes with index i > 2, and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 3, 2, 9, 19, 6, 61, 27, 4, 57, 11, 18, 281, 183, 38, 81, 101, 12, 5, 171, 122, 33, 263, 54, 361, 843, 8, 549, 29, 114, 59, 243, 22, 303, 1159, 36, 1811, 15, 562, 513, 1091, 366, 157, 99, 76, 789, 409, 162, 3721, 1083, 202, 2529, 541, 24, 209, 1647, 10, 87, 31

Offset: 0

Antti Karttunen, Dec 31 2013

The permutation satisfies A008578(a(n)) = a(A008578(n)) for all n, and is self-inverse.
The sequence of fixed points begins as 0, 1, 6, 11, 29, 36, 66, 95, 107, 121, 149, 174, 216, 313, 319, 396, 427, ... and is itself multiplicative in a sense that if a and b are fixed points, then also a*b is a fixed point.
The records are 0, 1, 3, 9, 19, 61, 281, 361, 843, 1159, 1811, 3721, 5339, 5433, 17141, 78961, 110471, 236883, 325679, ...
and they occur at positions 0, 1, 2, 4, 5, 7, 13, 25, 26, 35, 37, 49, 65, 74, 91, 169, 259, 338, 455, ...
(Note how the permutations map squares to squares, and in general keep the prime signature the same.)
Composition with similarly constructed A235199 gives the permutations A234743 & A234744 with more open cycle-structure.
The result of applying a permutation of the prime numbers to the prime factors of n. - Peter Munn, Dec 15 2019

			a(4) = a(2 * 2) = a(2)*a(2) = 3*3 = 9.
a(5) = a(p_3) = p_{a(3+1)-1} = p_{9-1} = p_8 = 19.
a(11) = a(p_5) = p_{a(5+1)-1} = p_{a(6)-1} = p_5 = 11.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences that are permutations of the natural numbers

List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property. Apart from the first one, all satisfy A000040(a(n)) = a(A000040(n)) except for a finite number of cases (with A235200, substitute A065091 for A000040):
A235200 (swaps 3 & 5).
A235199 (swaps 5 & 7).
A235201 (swaps 3 & 4).
A235487 (swaps 7 & 8).
A235489 (swaps 8 & 9).
Cf. A064614, A234743/A234744, A235485/A235486, A235493/A235494, also A317930.
Properties preserved by the sequence as a function: A000005, A001221, A001222, A051903, A101296.
A007913, A007947, A008578, A019554, A055231, A059895, A059896, A059897 are used to express relationships between terms of this sequence.

a[n_] := a[n] = Switch[n, 0, 0, 1, 1, 2, 3, 3, 2, _, Product[{p, e} = pe; Prime[a[PrimePi[p] + 1] - 1]^e, {pe, FactorInteger[n]}]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 21 2021 *)

A234840(n) = if(n<=1,n,my(f = factor(n)); for(i=1, #f~, if(2==f[i,1], f[i,1]++, if(3==f[i,1], f[i,1]--, f[i,1] = prime(-1+A234840(1+primepi(f[i,1])))))); factorback(f)); \\ Antti Karttunen, Aug 23 2018

a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 2, a(p_i) = p_{a(i+1)-1} for primes with index i > 2, and a(u * v) = a(u) * a(v) for u, v > 0.
From Peter Munn, Dec 14 2019. These identities would hold also if a(n) applied any other permutation of the prime numbers to the prime factors of n: (Start)
A000005(a(n)) = A000005(n).
A001221(a(n)) = A001221(n).
A001222(a(n)) = A001222(n).
A051903(a(n)) = A051903(n).
A101296(a(n)) = A101296(n).
a(A007913(n)) = A007913(a(n)).
a(A007947(n)) = A007947(a(n)).
a(A019554(n)) = A019554(a(n)).
a(A055231(n)) = A055231(a(n)).
a(A059895(n,k)) = A059895(a(n), a(k)).
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A059897(n,k)) = A059897(a(n), a(k)).
(End)

A235201 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=4 and a(4)=3, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 8, 5, 6, 16, 14, 17, 12, 19, 10, 28, 9, 11, 32, 13, 21, 20, 34, 53, 24, 49, 38, 64, 15, 43, 56, 59, 18, 68, 22, 35, 48, 37, 26, 76, 42, 67, 40, 29, 51, 112, 106, 107, 36, 25, 98, 44, 57, 23, 128, 119, 30, 52, 86, 31, 84, 131, 118, 80, 27, 133, 136, 41, 33

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=2, and is self-inverse. It swaps 3 & 4, maps any prime p_i with index i > 2 to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection on non-oriented rooted trees, mapped through the Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 3 and 4, wherever they occur as the terminal configurations anywhere in the tree:
....................
.o..................
.|..................
.o.............o...o
.|..............\./.
.x.....<--->.....x..
.3...............4..
That is, the last two edges of any branch which ends with at least in two edges long unbranched stem, will be changed to a V-branch (two single edges in parallel). Vice versa, any terminal configuration in the tree that consists of more than one single edges next to each other (in "parallel") will be transformed so that maximal even number (2k) of those single edges will be combined to k unbranching stems of two edges, and an extra odd edge, if present, will stay as it is.
This permutation commutes with A235199, i.e. a(A235199(n)) = A235199(a(n)) for all n. This can be easily seen, when comparing the above bijection to the one described in A235199. Composition A235199 o A235201 works as a "difference" of these two bijections, swapping the above subconfigurations only when they do not occur alone at the tips of singular edges. (Which cases are encoded by Matula-Goebel numbers 5 and 7, the third and fourth prime respectively).
Permutation fixes n! for n=0, 1, 2, 4, 7.
Note that a(5!) = a(120) = 168 = 120+(2*4!) and a(8!) = a(40320) = 30240 = 40320-(2*7!).

Composition with A235487 gives A235485/A235486, composition with A235489 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A072026, A061773.

Multiplicative with a(3^k) = 2^(2k), a(2^(2k)) = 3^k, a(2^(2k+1)) = 2*3^k, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 17, 12, 13, 10, 21, 16, 11, 18, 19, 28, 15, 34, 23, 24, 49, 26, 27, 20, 43, 42, 59, 32, 51, 22, 35, 36, 37, 38, 39, 56, 41, 30, 29, 68, 63, 46, 73, 48, 25, 98, 33, 52, 53, 54, 119, 40, 57, 86, 31, 84, 61, 118, 45, 64, 91, 102

Offset: 0

Antti Karttunen, Jan 04 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=3 or 4, and is self-inverse. It swaps 5 & 7, maps all larger primes p_i (with index i > 4) to p_{a(i)}, and lets the multiplicativity take care of the rest.
It can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). The bijection will swap the subtrees encoded by primes 5 and 7, wherever they occur as the terminal branches of the tree:
....................
.o..................
.|..................
.o.............o...o
.|..............\./.
.o.....<--->.....o..
.|...............|..
.x...............x..
.5...............7..
That is, any branch which ends at least in three edges long unbranched stem, will be changed so that its last two edges will become V-branch. Vice versa, any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will be straightened to an unbranching stem of three edges.
This permutation commutes with A235201, i.e. a(A235201(n)) = A235201(a(n)) for all n.
Permutation fixes n! for n=0, 1, 2, 3, 4, 7, 8 and 9.
Note also that a(5!) = a(120) = 168 = 120+(2*4!) and a(10!) = 5080320 = 3628800+(4*9!).

Composition with A234840 gives A234743 & A234744.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A235485/A235486, A235493/A235494, A091204/A091205, A072026, A061773.

For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 18, 17, 16, 23, 20, 21, 22, 19, 27, 25, 26, 24, 28, 29, 30, 31, 36, 33, 34, 35, 32, 37, 46, 39, 45, 41, 42, 43, 44, 40, 38, 47, 54, 49, 50, 51, 52, 61, 48, 55, 63, 69, 58, 59, 60, 53, 62, 56, 81, 65, 66, 83, 68, 57, 70, 71, 72, 73, 74, 75, 92, 77, 78, 79, 90, 64

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n, and is self-inverse. It swaps 8 & 9, maps any prime p_i with index i to p_{a(i)}, and lets the multiplicativity take care of the rest.
This can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 8 and 9, wherever they occur as the terminal branches of the tree:
.......................
.................o...o.
.................|...|.
.o.o.o...........o...o.
..\|/.............\./..
...x.....<--->.....x...
...8...............9...
Thus, any terminal configuration in the tree that consists of three or more single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be replaced by k subtrees Matula-Goebel-encoded by 9 (see above, or equally: replaced by 2k two-edges-long branches encoded by 3), and one or two left-over single edges, if present, will stay as they are. Vice versa, any terminal configuration in the tree that consists of more than one two-edges-long branches next to each other (in "parallel") will be transformed so that maximal even number (2k) of those double-edges will be replaced by 3k single edges, and an extra odd double-edge, if present, will stay as it is.
Note how in contrast to A235487, A235201 and A235199, this bijection is not size-preserving (the number of edges will change), which has implications when composing this with other such permutations (cf. e.g. A235493/A235494).

Composition with A235201 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(3^(2k)) = 2^3k = 8^k, a(3^(2k+1)) = 3*2^3k, a(2^(3k)) = 3^2k = 9^k, a(2^(3k+1)) = 2*9^k, a(2^(3k+2)) = 4*9^k, a(p_i) = p_{a(i)} for primes with index i, and a(u*v) = a(u) * a(v) for composites other than 8 or 9.

A235200 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=5 and a(5)=3, a(p_i) = p_{a(i-1)+1} for primes with index i > 3, and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 11, 20, 7, 26, 15, 16, 31, 50, 43, 12, 65, 22, 23, 40, 9, 14, 125, 52, 101, 30, 17, 32, 55, 62, 39, 100, 37, 86, 35, 24, 73, 130, 19, 44, 75, 46, 103, 80, 169, 18, 155, 28, 53, 250, 33, 104, 215, 202, 59, 60, 131, 34, 325, 64

Offset: 0

Antti Karttunen, Jan 04 2014

The permutation satisfies A065091(a(n)) = a(A065091(n)) for all n >= 3, and is self-inverse: It swaps 3 & 5, maps all larger primes p_i (with index i > 3) to p_{a(i-1)+1}, and lets the multiplicativity take care of the rest.

Antti Karttunen, Table of n, a(n) for n = 0..5520
Index entries for sequences that are permutations of the natural numbers

List below gives similarly constructed permutations, which all force a swap of two small numbers, with the composite numbers defined by the multiplicative property and the fact that (the rest of) primes are permuted with the sequence itself. Apart from the first one, all satisfy A000040(a(n)) = a(A000040(n)) except for a finite number of cases (with A234840, substitute A008578 for A000040):
A234840 (swaps 2 & 3).
A235199 (swaps 5 & 7).
A235201 (swaps 3 & 4).
A235487 (swaps 7 & 8).
A235489 (swaps 8 & 9).
Cf. also A072029, A181351, A234743/A234744.

For n < 3, a(n)=n, a(3)=5 and a(5)=3, a(p_i) = p_{a(i-1)+1} for primes with index i > 3, and a(u * v) = a(u) * a(v) for u, v > 0.

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

cp's OEIS Frontend

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

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A235486 Permutation of natural numbers: a(n) = A235487(A235201(n)).

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A234840 Self-inverse and multiplicative permutation of integers: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 2, a(p_i) = p_{a(i+1)-1} for primes with index i > 2, and a(u * v) = a(u) * a(v) for u, v > 0.

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A235201 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=4 and a(4)=3, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

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A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(u*v) = a(u) * a(v).

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A235200 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=5 and a(5)=3, a(p_i) = p_{a(i-1)+1} for primes with index i > 3, and a(u * v) = a(u) * a(v) for u, v > 0.

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).