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A354225 Lexicographically earliest sequence of distinct positive integers such that a(1) = 1 and for any n > 1, n / gcd(n, a(n)) and a(n) / gcd(n, a(n)) are prime.

%I A354225 #13 May 22 2022 14:05:46
%S A354225 1,3,2,6,7,4,5,12,15,14,13,8,11,10,9,24,19,27,17,28,33,26,29,16,35,22,
%T A354225 18,20,23,42,37,48,21,38,25,54,31,34,51,56,43,30,41,52,63,58,53,32,77,
%U A354225 70,39,44,47,36,65,40,69,46,61,84,59,74,45,96,55,78,71
%N A354225 Lexicographically earliest sequence of distinct positive integers such that a(1) = 1 and for any n > 1, n / gcd(n, a(n)) and a(n) / gcd(n, a(n)) are prime.
%C A354225 This sequence is a self-inverse permutation of the positive integers that preserves the number of prime divisors (with or without multiplicity).
%H A354225 Michael De Vlieger, <a href="/A354225/b354225.txt">Table of n, a(n) for n = 1..10000</a>
%H A354225 Michael De Vlieger, <a href="/A354225/a354225.png">Annotated log-log plot of a(n)</a>, n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green, fixed points in gold, and composite prime powers in magenta.
%H A354225 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A354225 a(prime(2*n)) = prime(2*n-1) (where prime(n) denotes the n-th prime number).
%e A354225 The first terms are:
%e A354225   n   a(n)  g=gcd(n, a(n))  n/g  a(n)/g
%e A354225   --  ----  --------------  ---  ------
%e A354225    1     1               1    1       1
%e A354225    2     3               1    2       3
%e A354225    3     2               1    3       2
%e A354225    4     6               2    2       3
%e A354225    5     7               1    5       7
%e A354225    6     4               2    3       2
%e A354225    7     5               1    7       5
%e A354225    8    12               4    2       3
%e A354225    9    15               3    3       5
%e A354225   10    14               2    5       7
%e A354225   11    13               1   11      13
%e A354225   12     8               4    3       2
%e A354225   13    11               1   13      11
%e A354225   14    10               2    7       5
%t A354225 nn = 120; c[_] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[{i/#, k/#}, PrimeQ] &@ GCD[i, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, May 22 2022 *)
%o A354225 (PARI) s=0; for (n=1, 67, for (v=1, oo, if (!bittest(s, v) && (n==1 || (isprime(n/g=gcd(n,v)) && isprime(v/g))), print1 (v", "); s+=2^v; break)))
%Y A354225 Cf. A122280.
%K A354225 nonn
%O A354225 1,2
%A A354225 _Rémy Sigrist_, May 20 2022