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A366807 a(n) = A020639(A120944(n))*A120944(n).

%I A366807 #19 Aug 02 2024 21:05:43
%S A366807 12,20,28,45,63,44,52,60,99,68,175,76,117,84,92,153,275,171,116,124,
%T A366807 325,132,207,140,148,539,156,164,425,172,261,637,279,188,475,204,315,
%U A366807 212,220,333,228,575,236,833,244,369,387,260,931,268,276,423,284,1573,725
%N A366807 a(n) = A020639(A120944(n))*A120944(n).
%C A366807 Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across composite squarefree numbers A120944.
%C A366807 Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k. It is plain to see that k is the first term in the sequence k*R_k. This sequence gives the second term in k*R_k since lpf(k) is the second term in R_k.
%C A366807 Permutation of A366825. Contains numbers whose prime signature has at least 2 terms, of which is 2, the rest of which are 1s.
%C A366807 Proper subset of A364996, which itself is contained in A126706.
%H A366807 Michael De Vlieger, <a href="/A366807/b366807.txt">Table of n, a(n) for n = 1..10000</a>
%F A366807 a(n) = A065642(A120944(n)), n > 1.
%F A366807 a(n) = A285109(A120944(n)).
%e A366807 Let b(n) = A120944(n).
%e A366807 a(1) = 12 = 2^2*3^1 = b(1)*lpf(b(1)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term.
%e A366807 a(2) = 20 = 2^2*5^1 = b(2)*lpf(b(2)) = 10*lpf(10) = 10*2. In {10*A003592}, 20 is the second term.
%e A366807 a(4) = 45 = 3^2*5^1 = b(4)*lpf(b(4)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.
%t A366807 nn = 150; s = Select[Range[nn], And[SquareFreeQ[#], CompositeQ[#]] &];
%t A366807 Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]
%o A366807 (Python)
%o A366807 from math import isqrt
%o A366807 from sympy import primepi, mobius, primefactors
%o A366807 def A366807(n):
%o A366807     def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A366807     m, k = n+1, f(n+1)
%o A366807     while m != k:
%o A366807         m, k = k, f(k)
%o A366807     return m*min(primefactors(m)) # _Chai Wah Wu_, Aug 02 2024
%Y A366807 Cf. A000040, A007947, A020639, A065642, A120944, A126706, A285109, A364996, A366825.
%K A366807 nonn,easy
%O A366807 1,1
%A A366807 _Michael De Vlieger_, Dec 16 2023