A235489 -id:A235489 - cp's OEIS frontend

A235493 Permutation of natural numbers: a(n) = A235201(A235489(n)).

0, 1, 2, 4, 3, 7, 8, 5, 16, 6, 14, 17, 12, 19, 10, 28, 32, 11, 9, 53, 21, 20, 34, 13, 64, 49, 38, 24, 15, 43, 56, 59, 48, 68, 22, 35, 18, 37, 106, 76, 112, 67, 40, 29, 51, 42, 26, 107, 128, 25, 98, 44, 57, 131, 36, 119, 80, 212, 86, 31, 84, 23, 118, 30, 256

Offset: 0

Antti Karttunen, Jan 11 2014

nonn

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except for n=2.

In contrast to A235485/A235486 this permutation also contains infinite cycles.

Antti Karttunen, Table of n, a(n) for n = 0..9856
Index entries for sequences that are permutations of the natural numbers

Inverse: A235494. Cf. also A235201, A235489, A235485/A235486, A234743/A234744.

(define (A235493 n) (A235201 (A235489 n)))

a(n) = A235201(A235489(n)).

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

A235494 Permutation of natural numbers: a(n) = A235489(A235201(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 9, 5, 6, 18, 14, 17, 12, 23, 10, 28, 8, 11, 36, 13, 21, 20, 34, 61, 27, 49, 46, 81, 15, 43, 63, 59, 16, 68, 22, 35, 54, 37, 26, 92, 42, 83, 45, 29, 51, 126, 122, 107, 32, 25, 98, 44, 69, 19, 162, 119, 30, 52, 86, 31, 84, 151, 118, 90, 24

Offset: 0

Antti Karttunen, Jan 11 2014

nonn

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except for n=2.

In contrast to A235485/A235486 this permutation contains also infinite cycles.

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

Inverse: A235493. Cf. also A235201, A235489, A235485/A235486, A234743/A234744.

(define (A235494 n) (A235489 (A235201 n)))

a(n) = A235489(A235201(n)).

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

A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Henry Bottomley, Dec 31 2002

nonn

Fernandez calls this the order of primeness of n.
a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013
When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014
Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024

			a(1) = 0 since 1 is not prime;
a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
a(4) = 0 since 4 is not prime;
a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
a(6) = 0 since 6 is not prime;
a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Index entries for sequences related to Matula-Goebel numbers

A left inverse of A007097.
One less than A049076.
a(A000040(n)) = A049076(n).
Cf. A373338 (mod 2), A018252 (positions of zeros).
Cf. A000720, A049084, A061773.
Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).
Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.

a078442 n = fst $ until ((== 0) . snd)
                        (\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
-- Reinhard Zumkeller, Jul 14 2013

A078442 := proc(n)
    if not isprime(n) then
        0 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc: # R. J. Mathar, Jul 07 2012

a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* Jean-François Alcover, Jan 26 2018 *)

A078442(n)=for(i=0,n, isprime(n) || return(i); n=primepi(n)) \\ M. F. Hasler, Mar 09 2010

a(n) = A049076(n)-1.
a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - Reinhard Zumkeller, Jul 14 2013
For all n, a(n) = A007814(A135141(n)) and a(A227413(n)) = A007814(n). Also a(A235489(n)) = a(n). - Antti Karttunen, Jan 27 2014

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]

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 16, 15, 14, 19, 18, 17, 20, 24, 22, 23, 21, 25, 26, 27, 32, 29, 30, 31, 28, 33, 38, 40, 36, 37, 34, 39, 35, 41, 48, 53, 44, 45, 46, 47, 42, 64, 50, 57, 52, 43, 54, 55, 56, 51, 58, 67, 60, 61, 62, 72, 49, 65, 66, 59, 76, 69, 80, 71, 63, 89, 74, 75, 68, 88, 78, 79, 70, 81

Offset: 0

Antti Karttunen, Jan 11 2014

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=4, and is self-inverse. It swaps 7 & 8, maps any prime p_i with index i > 4 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 7 and 8, wherever they occur as the terminal configurations anywhere in the tree:
.......................
.o...o.................
..\./..................
...o.............o.o.o.
...|..............\|/..
...x.....<--->.....x...
...7...............8...
Thus any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will emanate "in parallel" from the same vertex. Vice versa, any terminal configuration in the tree that consists of more than two single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be transformed to k Y-formations, and one or two left-over edges, if present, will stay as they are.

Composition with A235201 gives A235485/A235486.
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).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(p_i) = p_{a(i)} for primes with index i <> 4, a(7) = 8, a(2^(3k)) = 7^k, a(2^(3k+1)) = 2*7^k, a(2^(3k+2)) = 4*7^k, and for other composites, a(u * v) = a(u) * a(v).

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

A235493 Permutation of natural numbers: a(n) = A235201(A235489(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

Extensions

A235494 Permutation of natural numbers: a(n) = A235489(A235201(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

Extensions

A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(u*v) = a(u) * a(v).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).