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

Author

Antti Karttunen, Dec 31 2013

Keywords

• nonn
• mult

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

Examples

			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.
		

Links

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

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.

Programs

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 *)

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

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)