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).

18, 24, 36, 48, 50, 54, 72, 75, 80, 90, 96, 98, 100, 108, 112, 120, 126, 135, 144, 147, 150, 160, 162, 168, 180, 189, 192, 196, 198, 200, 216, 224, 225, 234, 240, 242, 245, 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

Offset: 1

Author

Michael De Vlieger, Feb 22 2023

Keywords

• nonn

Comments

Proper subsequence of A126706.

Numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

Examples

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. Note: rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. Note: rad(24) = rad(36) = 6 and 36 mod 24 = 12.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, 1016 pixel square bitmap 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)
• Michael De Vlieger, 1016 pixel square bitmap as described above, at scale 1X.

Crossrefs

Cf. A007947, A013929, A024619, A119288, A126706, A355432.

Programs

Mathematica

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

Formula

This sequence is { k in A126706 : k/A007947(k) >= A119288(k) }.