|
%I #7 Jan 24 2024 10:13:08
%S 36,72,108,144,216,288,324,432,576,648,864,900,972,1152,1296,1728,
%T 1800,1944,2304,2592,2700,2916,3456,3600,3888,4500,4608,5184,5400,
%U 5832,6912,7200,7776,8100,8748,9000,9216,10368,10800,11664,13500,13824,14400,15552,16200
%N Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.
%C Numbers k such that Omega(k) > omega(k) > 1 with all prime power factors p^m for m > 1, such that squarefree kernel rad(k) is in A002110, where Omega = A001222, omega = A001221, and rad(k) = A007947(k).
%C Union of the product of the squares of primorials P(n)^2, n > 1, and the set of prime(n)-smooth numbers.
%C Superset of A364930.
%C Proper subset of A367268, which in turn is a proper subset of A126706.
%H Michael De Vlieger, <a href="/A369374/b369374.txt">Table of n, a(n) for n = 1..10000</a>
%F {a(n)} = { m*P(n)^2 : P(n) = Product_{j = 1..n} prime(n), rad(m) | P(n), n > 1 }.
%F Intersection of A286708 and A055932.
%F A286708 is the union of A369417 and this sequence.
%e This sequence is the union of the following infinite sets:
%e P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
%e = { m*P(2)^2 : rad(m) | P(2) }.
%e P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
%e = { m*P(3)^2 : rad(m) | P(3) }.
%e P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
%e = { m*P(4)^2 : rad(m) | P(4) }, etc.
%t With[{nn = 2^14},
%t Select[
%t Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
%t Not@*PrimePowerQ],
%t And[EvenQ[#],
%t Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]
%Y Cf. A001221, A001222, A001694, A002110, A007947, A055932, A126706, A286708, A364930, A367268, A369417.
%K nonn
%O 1,1
%A _Michael De Vlieger_, Jan 22 2024
|
