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A358971 a(1) = 1. Thereafter a(n) is least novel k != n such that rad(k) = rad(n), where rad is A007947.

%I A358971 #25 Feb 12 2023 17:27:36
%S A358971 1,4,9,2,25,12,49,16,3,20,121,6,169,28,45,8,289,24,361,10,63,44,529,
%T A358971 18,5,52,81,14,841,60,961,64,99,68,175,48,1369,76,117,50,1681,84,1849,
%U A358971 22,15,92,2209,36,7,40,153,26,2809,72,275,98,171,116,3481,30,3721
%N A358971 a(1) = 1. Thereafter a(n) is least novel k != n such that rad(k) = rad(n), where rad is A007947.
%C A358971 In other words, for n > 1, a(n) is the least novel k other than n which has not occurred earlier whose squarefree kernel is equal to the squarefree kernel of n.
%C A358971 Conjectured to be a permutation of the positive integers with primes appearing in natural order. Primes are minima, 1 and primes squared are records.
%C A358971 From _Michael De Vlieger_, Dec 07 2022: (Start)
%C A358971 Some consequences of definition:
%C A358971 There are no fixed points outside of a(1) = 1.
%C A358971 Prime power p^e implies a(p^e) = p^(e+1) for odd e, else p^(e-1). Hence a(p) = p^2 comprise maxima, while a(p^2) = p comprise minima.
%C A358971 Let lpf(m) = least prime factor of m. Squarefree m implies a(m) = lpf(m)*m and a(lpf(m)*m) = m, as seen in scatterplot in rays with slope p and 1/p, respectively. Therefore squarefree numbers are sequestered along or below a(n/2) = n/2.
%C A358971 Let K = rad(n); a(n) and n (such that a(n) != n) belong to the same sequence K*R_K, where R_K is the list of K-regular numbers, 1 and those whose prime divisors are restricted to p | K. For example, if K = 6, then a(n) and n belong to 6*A003586, and if K = 10, then a(n) and n belong to 10*A003592.
%C A358971 Observation: For m in A286708, abs(a(m) - m) is relatively small. (End)
%C A358971 This sequence is a self-inverse permutation of the positive integers: for any squarefree number s > 1, let v_s be the list of numbers with radical s, then for any k > 0, a(v_s(2*k)) = v_s(2*k-1) and a(v_s(2*k-1)) = v_s(2*k). - _Rémy Sigrist_, Dec 08 2022
%H A358971 Michael De Vlieger, <a href="/A358971/b358971.txt">Table of n, a(n) for n = 1..10000</a>
%H A358971 Michael De Vlieger, <a href="/A358971/a358971.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A358971 Michael De Vlieger, <a href="/A358971/a358971_1.png">Log log scatterplot of a(n)</a>, n = 1..2^14, highlighting primes in red, composite prime powers (in A246547) in gold, composite squarefree numbers (A120944) in green, numbers neither squarefree nor prime power (in A126706) in blue, with numbers in A286706 in large light blue. Gold and light blue numbers are in A001694. Maxima are a(p) = p^2, minima are a(p^2) = p.
%H A358971 Michael De Vlieger, <a href="/A358971/a358971_2.png">Log log scatterplot of a(n)</a>, n = 1..2^10, using the same color coding as immediately above, labeled and showing quasi-rays with slopes p and 1/p.
%H A358971 Michael De Vlieger, <a href="/A358971/a358971_3.png">Log log scatterplot of a(n)</a> n = 1..2^16, showing n | a(n) in green, a(n) | n in red, and other terms in blue.
%H A358971 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A358971 For squarefree n, a(a(n)) = n; a(p) = p^2 for p prime, and a(p^2) = p.
%e A358971 a(2) = 4 because 4 is the least number (not equal to 2) which has the same squarefree kernel as 2.
%e A358971 a(4) = 2 because 2 is the least unused number (not equal to 4) having the same squarefree kernel as 4
%t A358971 nn = 61; c[_] = False; q[_] = 1; f[n_] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], True, k = f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, f[q[k]]]], q[k]++]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Dec 07 2022 *)
%Y A358971 Cf. A000040, A001248, A005117, A007947, A253288, A358916.
%K A358971 nonn
%O A358971 1,2
%A A358971 _David James Sycamore_, Dec 07 2022
%E A358971 More terms from _Michael De Vlieger_, Dec 07 2022