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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. 12
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 (list; graph; refs; listen; history; text; internal format)
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.

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.

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.

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. A008578, A064614, A234743/A234744, A235485/A235486, A235493/A235494, also A317930.

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

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Last modified February 19 08:57 EST 2019. Contains 320309 sequences. (Running on oeis4.)