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.

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

0,3

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 0..16384

Index entries for sequences that are permutations of the natural numbers

Formula

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)

Example

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.

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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.

Sequence in context: A049969 A088634 A118791 * A234743 A284989 A049971

Adjacent sequences: A234837 A234838 A234839 * A234841 A234842 A234843

Keyword

nonn,mult

Author

Antti Karttunen, Dec 31 2013

Status

approved