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A358786 a(1) = 1. For n > 1, a(n) is least novel k != n such that rad(k) = rad(n) and either k | n or n | k, where rad is A007947.

%I A358786 #7 Dec 11 2022 10:32:17
%S A358786 1,4,9,2,25,12,49,16,3,20,121,6,169,28,45,8,289,36,361,10,63,44,529,
%T A358786 48,5,52,81,14,841,60,961,64,99,68,175,18,1369,76,117,80,1681,84,1849,
%U A358786 22,15,92,2209,24,7,100,153,26,2809,108,275,112,171,116,3481,30,3721
%N A358786 a(1) = 1. For n > 1, a(n) is least novel k != n such that rad(k) = rad(n) and either k | n or n | k, where rad is A007947.
%C A358786 Variant of A358971 that additionally requires either k | n or n | k. This version eliminates nondivisor n and a(n) seen in a scatterplot of A358971. First differs from A358971 at n = 18.
%C A358786 Some consequences of definition:
%C A358786 There are no fixed points outside of a(1) = 1.
%C A358786 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 A358786 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 A358786 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.
%H A358786 Michael De Vlieger, <a href="/A358786/a358786.png">Log log scatterplot of a(n)</a> n = 1..2^20.
%H A358786 Michael De Vlieger, <a href="/A358786/a358786_1.png">Log log scatterplot of a(n)</a> n = 1..2^10, showing primes in red, composite prime powers (in A246547) in gold, squarefree composites (in A120944) in green, numbers neither squarefree nor prime power (in A126706) in blue, highlighting numbers in A286708 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 A358786 Michael De Vlieger, <a href="/A358786/a358786_2.png">Log log scatterplot of a(n)</a>, n = 1..2^16, showing n | a(n) in green, a(n) | n in red.
%t A358786 nn = 61; c[_] = False; q[_] = 1; f[n_] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; Do[Which[PrimePowerQ[n], k = If[OddQ[#2], #1^(#2 + 1), #1^(#2 - 1)] & @@ First@ FactorInteger[n], PrimeQ@ Sqrt[n], k = Sqrt[n], True, k = f[n]; m = q[k]; While[Nand[! c[k m], Or[Divisible[k m, n], Divisible[n, 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}], {n, 2, nn}]; Array[a, nn]
%Y A358786 Cf. A007947, A358971.
%K A358786 nonn
%O A358786 1,2
%A A358786 _Michael De Vlieger_, Dec 08 2022