cp's OEIS Frontend

A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).

%I A369417 #5 Jan 24 2024 10:13:53
%S A369417 100,196,200,225,392,400,441,484,500,675,676,784,800,968,1000,1089,
%T A369417 1125,1156,1225,1323,1352,1372,1444,1521,1568,1600,1764,1936,2000,
%U A369417 2025,2116,2312,2500,2601,2704,2744,2888,3025,3087,3136,3200,3249,3267,3364,3375,3528
%N A369417 Powerful numbers k with multiple distinct prime factors such that rad(k) is not a primorial, where rad(k) = A007947(k).
%C A369417 Numbers k such that Omega(k) > omega(k) > 1, where all prime power factors p^m have exponents m > 1, such that squarefree kernel rad(k) not in A002110, where Omega = A001222 and omega = A001221.
%H A369417 Michael De Vlieger, <a href="/A369417/b369417.txt">Table of n, a(n) for n = 1..10000</a>
%F A369417 {a(n)} = { A286708 \ A055932 }.
%F A369417 {a(n)} = { m*s^2 : Omega(s) = omega(s) > 1, s not in A002110, rad(m) | s }.
%F A369417 A286708 is the union of A369374 and this sequence.
%e A369417 Let S = A366413 = {A120944 \ A002110}.
%e A369417 This sequence is the union of the following infinite sets:
%e A369417 S(1)^2 * A003592 = 10^2 * A003592 = {100, 200, 400, 500, 800, 1000, ...}
%e A369417                  = { m*S(1)^2 : rad(m) | S(1) }.
%e A369417 S(2)^2 * A003591 = 14^2 * A003591 = {196, 392, 784, 1372, 1568, ...}
%e A369417                  = { m*S(2)^2 : rad(m) | S(2) }.
%e A369417 S(3)^2 * A003593 = 15^2 * A003593 = {225, 675, 1125, 2025, 3375, ...}
%e A369417                  = { m*S(3)^2 : rad(m) | S(3) }, etc.
%t A369417 With[{nn = 2^14},
%t A369417   Select[
%t A369417     Select[
%t A369417       Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
%t A369417     Not@*PrimePowerQ],
%t A369417   Nand[EvenQ[#],
%t A369417     Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]
%Y A369417 Cf. A001221, A001222, A002110, A055932, A120944, A126706, A286708, A366413, A369374.
%K A369417 nonn
%O A369417 1,1
%A A369417 _Michael De Vlieger_, Jan 22 2024