A372404 - OEIS

%I #7 Jun 07 2024 14:21:47

%S 72,108,144,200,216,288,324,392,400,432,500,576,648,675,784,800,864,

%T 968,972,1000,1125,1152,1296,1323,1352,1372,1568,1600,1728,1800,1936,

%U 1944,2000,2025,2304,2312,2500,2592,2700,2704,2744,2888,2916,3087,3136,3200,3267,3375,3456

%N Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

%C A001694 \ A246547 = A286708, i.e., A286708 contains powerful numbers without perfect prime powers. Hence, this sequence is a proper subset of A286708 which in turn is contained in A126706.

%C Numbers k in A286708 are such that rad(k)^2 | k. Numbers in this sequence are such that k != A120944(m)^2 for some m, where A120944 is the sequence of squarefree composites.

%H Michael De Vlieger, <a href="/A372404/b372404.txt">Table of n, a(n) for n = 1..10000</a>

%F A286708 = union of A177492 and this sequence.

%F A001694 = union of A246547, A177492, and this sequence.

%F A126706 = union of A332785, A177492, and this sequence.

%e The number 36 is not in the sequence since 36/rad(36) = 36/6 = 6, squarefree.

%e a(1) = 72 since 72/rad(72) = 72/6 = 12 is nonsquarefree.

%e a(2) = 108 since 108/rad(108) = 108/6 = 18 is nonsquarefree.

%e a(4) = 200 since 200/rad(200) = 200/10 = 20 is nonsquarefree, etc.

%t With[{nn = 3300},

%t   Select[

%t     Select[Rest@ Union@ Flatten@

%t       Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],

%t     Not@*PrimePowerQ],

%t   Not@ SquareFreeQ[#/(Times @@ FactorInteger[#][[;;, 1]])] &] ]

%o (PARI) rad(n) = factorback(factorint(n)[, 1]);

%o isok(k) = ispowerful(k) && !isprimepower(k) && !issquarefree(k/rad(k)); \\ _Michel Marcus_, Jun 05 2024

%Y Cf. A001694, A007947, A013929, A120944, A126708, A177492, A126708, A246547, A286708, A332785.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jun 04 2024