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A363082 Numbers k neither squarefree nor prime power such that q*r > k, where q = A053669(k) is the smallest prime that does not divide k and r = A007947(k) is the squarefree kernel.

%I A363082 #48 Aug 02 2023 13:55:14
%S A363082 12,18,20,24,28,44,52,60,68,76,84,90,92,116,120,124,126,132,140,148,
%T A363082 150,156,164,168,172,180,188,198,204,212,220,228,234,236,244,260,264,
%U A363082 268,276,284,292,306,308,312,316,332,340,342,348,356,364,372,380,388,404,408,412,414,420
%N A363082 Numbers k neither squarefree nor prime power such that q*r > k, where q = A053669(k) is the smallest prime that does not divide k and r = A007947(k) is the squarefree kernel.
%H A363082 Michael De Vlieger, <a href="/A363082/b363082.txt">Table of n, a(n) for n = 1..10000</a>
%H A363082 Michael De Vlieger, <a href="/A363082/a363082.png">Plot b(n) = A126706(n) at (x, y)</a> for n = ym + x = 1..1032256, m = 1016 and x = 1..m, y = 0..m-1, showing b(n) in A360765 in white, and b(n) in this sequence in other colors, where red indicates b(n) also in A360767, and blue indicates b(n) also in A360768.
%F A363082 This sequence is A126706 \ A360765.
%e A363082 a(1) = 12 since 12 is the smallest number that is neither squarefree nor a prime power. Additionally, 12 < 5*6.
%e A363082 a(2) = 18 since it is in A126706, and like 12, 18 < 5*6.
%e A363082 a(3) = 20 since it is neither squarefree nor prime power, and 20 < 3*10.
%e A363082 36 is not in this sequence since 36 > 5*6.
%e A363082 40 is not in this sequence since 40 > 3*10, etc.
%t A363082 Select[Select[Range[452], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{q, r}, q r > k] @@ {SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]} ] @@ {#, FactorInteger[#]} &]
%Y A363082 Cf. A007947, A053669, A126706, A360765, A360767, A360768.
%K A363082 nonn
%O A363082 1,1
%A A363082 _Michael De Vlieger_, Jul 29 2023