cp's OEIS Frontend

A360768 Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

%I A360768 #16 Mar 04 2023 02:08:45
%S A360768 18,24,36,48,50,54,72,75,80,90,96,98,100,108,112,120,126,135,144,147,
%T A360768 150,160,162,168,180,189,192,196,198,200,216,224,225,234,240,242,245,
%U A360768 250,252,264,270,288,294,300,306,312,320,324,336,338,342,350,352,360,363,375,378,384,392,396,400,405,408
%N A360768 Numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).
%C A360768 Proper subsequence of A126706.
%C A360768 Numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.
%H A360768 Michael De Vlieger, <a href="/A360768/b360768.txt">Table of n, a(n) for n = 1..10000</a>
%H A360768 Michael De Vlieger, <a href="/A360768/a360768.png">1016 pixel square bitmap</a> of indices n = 1..1032256, read left to right, top to bottom, such that A126706(n) in this sequence appears in black and A126706(n) in A360767 in white. Shows a curious "sand ripple" pattern perhaps associated with congruence. (Magnification 3X)
%H A360768 Michael De Vlieger, <a href="/A360768/a360768_1.png">1016 pixel square bitmap</a> as described above, at scale 1X.
%F A360768 This sequence is { k in A126706 : k/A007947(k) >= A119288(k) }.
%e A360768 a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
%e A360768 a(2) = 24, since 24/6 >= 3. Note: rad(18) = rad(24) = 6 and 24 mod 18 = 6.
%e A360768 a(3) = 36, since 36/6 >= 3. Note: rad(24) = rad(36) = 6 and 36 mod 24 = 12.
%e A360768 a(6) = 54, since 54/6 >= 3. Note: m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.
%t A360768 Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]
%Y A360768 Cf. A007947, A013929, A024619, A119288, A126706, A355432.
%K A360768 nonn
%O A360768 1,1
%A A360768 _Michael De Vlieger_, Feb 22 2023