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%I #52 Nov 21 2021 12:34:42
%S 0,1,3,2,9,19,6,61,27,4,57,11,18,281,183,38,81,101,12,5,171,122,33,
%T 263,54,361,843,8,549,29,114,59,243,22,303,1159,36,1811,15,562,513,
%U 1091,366,157,99,76,789,409,162,3721,1083,202,2529,541,24,209,1647,10,87,31
%N 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.
%C The permutation satisfies A008578(a(n)) = a(A008578(n)) for all n, and is self-inverse.
%C 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.
%C The records are 0, 1, 3, 9, 19, 61, 281, 361, 843, 1159, 1811, 3721, 5339, 5433, 17141, 78961, 110471, 236883, 325679, ...
%C and they occur at positions 0, 1, 2, 4, 5, 7, 13, 25, 26, 35, 37, 49, 65, 74, 91, 169, 259, 338, 455, ...
%C (Note how the permutations map squares to squares, and in general keep the prime signature the same.)
%C Composition with similarly constructed A235199 gives the permutations A234743 & A234744 with more open cycle-structure.
%C The result of applying a permutation of the prime numbers to the prime factors of n. - _Peter Munn_, Dec 15 2019
%H Antti Karttunen, <a href="/A234840/b234840.txt">Table of n, a(n) for n = 0..16384</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F 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.
%F 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)
%F A000005(a(n)) = A000005(n).
%F A001221(a(n)) = A001221(n).
%F A001222(a(n)) = A001222(n).
%F A051903(a(n)) = A051903(n).
%F A101296(a(n)) = A101296(n).
%F a(A007913(n)) = A007913(a(n)).
%F a(A007947(n)) = A007947(a(n)).
%F a(A019554(n)) = A019554(a(n)).
%F a(A055231(n)) = A055231(a(n)).
%F a(A059895(n,k)) = A059895(a(n), a(k)).
%F a(A059896(n,k)) = A059896(a(n), a(k)).
%F a(A059897(n,k)) = A059897(a(n), a(k)).
%F (End)
%e a(4) = a(2 * 2) = a(2)*a(2) = 3*3 = 9.
%e a(5) = a(p_3) = p_{a(3+1)-1} = p_{9-1} = p_8 = 19.
%e a(11) = a(p_5) = p_{a(5+1)-1} = p_{a(6)-1} = p_5 = 11.
%t 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]}]];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 21 2021 *)
%o (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
%Y 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):
%Y A235200 (swaps 3 & 5).
%Y A235199 (swaps 5 & 7).
%Y A235201 (swaps 3 & 4).
%Y A235487 (swaps 7 & 8).
%Y A235489 (swaps 8 & 9).
%Y Cf. A064614, A234743/A234744, A235485/A235486, A235493/A235494, also A317930.
%Y Properties preserved by the sequence as a function: A000005, A001221, A001222, A051903, A101296.
%Y A007913, A007947, A008578, A019554, A055231, A059895, A059896, A059897 are used to express relationships between terms of this sequence.
%K nonn,mult
%O 0,3
%A _Antti Karttunen_, Dec 31 2013
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