cp's OEIS Frontend

A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

%I A372404 #7 Jun 07 2024 14:21:47
%S A372404 72,108,144,200,216,288,324,392,400,432,500,576,648,675,784,800,864,
%T A372404 968,972,1000,1125,1152,1296,1323,1352,1372,1568,1600,1728,1800,1936,
%U A372404 1944,2000,2025,2304,2312,2500,2592,2700,2704,2744,2888,2916,3087,3136,3200,3267,3375,3456
%N A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.
%C A372404 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 A372404 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 A372404 Michael De Vlieger, <a href="/A372404/b372404.txt">Table of n, a(n) for n = 1..10000</a>
%F A372404 A286708 = union of A177492 and this sequence.
%F A372404 A001694 = union of A246547, A177492, and this sequence.
%F A372404 A126706 = union of A332785, A177492, and this sequence.
%e A372404 The number 36 is not in the sequence since 36/rad(36) = 36/6 = 6, squarefree.
%e A372404 a(1) = 72 since 72/rad(72) = 72/6 = 12 is nonsquarefree.
%e A372404 a(2) = 108 since 108/rad(108) = 108/6 = 18 is nonsquarefree.
%e A372404 a(4) = 200 since 200/rad(200) = 200/10 = 20 is nonsquarefree, etc.
%t A372404 With[{nn = 3300},
%t A372404   Select[
%t A372404     Select[Rest@ Union@ Flatten@
%t A372404       Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
%t A372404     Not@*PrimePowerQ],
%t A372404   Not@ SquareFreeQ[#/(Times @@ FactorInteger[#][[;;, 1]])] &] ]
%o A372404 (PARI) rad(n) = factorback(factorint(n)[, 1]);
%o A372404 isok(k) = ispowerful(k) && !isprimepower(k) && !issquarefree(k/rad(k)); \\ _Michel Marcus_, Jun 05 2024
%Y A372404 Cf. A001694, A007947, A013929, A120944, A126708, A177492, A126708, A246547, A286708, A332785.
%K A372404 nonn,easy
%O A372404 1,1
%A A372404 _Michael De Vlieger_, Jun 04 2024