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A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

%I A367708 #14 Mar 02 2024 13:34:05
%S A367708 50,75,80,98,112,135,147,189,240,242,245,252,270,294,300,336,338,350,
%T A367708 352,360,363,378,396,416,450,468,480,490,504,507,525,528,540,550,560,
%U A367708 578,588,594,600,605,612,624,650,672,684,700,702,720,722,726,735,750,756
%N A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).
%C A367708 Does not contain 3-smooth numbers.
%C A367708 Contains neither A168263 nor A367511.
%C A367708 Conjecture: contains most highly composite numbers.
%H A367708 Michael De Vlieger, <a href="/A367708/b367708.txt">Table of n, a(n) for n = 1..10000</a>
%F A367708 Union of {k = m*s : rad(m) | s, max(p, q) <= m < s}, where s is in A120944.
%F A367708 {a(n)} = A364702 \ A366250.
%F A367708 {a(n)} = A361098 \ A341645.
%e A367708 Let q = A053669(k) and let p = A119288(k).
%e A367708 For s = 10, we have {50, 80}, since
%e A367708     s * { max(p, q) <= m < s  : rad(m) | s  }
%e A367708    = 10*{ max(5, 3) <= m < 10 : rad(m) | 10 }
%e A367708    = 10*{5, 8} = {50, 80}.
%e A367708 For s = 15, we have {45, 135}, since
%e A367708     s * { max(p, q) <= m < s  : rad(m) | s  }
%e A367708    = 15*{ max(5, 2) <= m < 15 : rad(m) | 15 }
%e A367708    = 15*{5, 9} = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.
%e A367708 For s = 30, we have {45, 135}, since
%e A367708     s * { max(p, q) <= m < s  : rad(m) | s  }
%e A367708    = 30*{ max(3, 7) <= m < 30 : rad(m) | 30 }
%e A367708    = 30*{8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27}
%e A367708    = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.
%t A367708 nn = 756;
%t A367708 Select[Select[Range[12, nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &],
%t A367708   And[Max[#2, #3] <= #1 < #4, ! AllTrue[#5, # > 1 &]] & @@
%t A367708     {#1/#4, #2, #3, #4, #5} & @@
%t A367708     {#1, #2[[2, 1]], #3, Times @@ #2[[All, 1]], #2[[All, -1]]} & @@
%t A367708     {#, FactorInteger[#], If[OddQ[#], 2,
%t A367708         q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]
%Y A367708 Cf. A002182, A003586, A053669, A119288, A120944, A168263, A341645, A361098, A364702, A366250, A367511.
%K A367708 nonn
%O A367708 1,1
%A A367708 _Michael De Vlieger_, Feb 09 2024