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

100, 196, 200, 225, 392, 400, 441, 484, 500, 675, 676, 784, 800, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1764, 1936, 2000, 2025, 2116, 2312, 2500, 2601, 2704, 2744, 2888, 3025, 3087, 3136, 3200, 3249, 3267, 3364, 3375, 3528

Offset: 1

Author

Michael De Vlieger, Jan 22 2024

Keywords

• nonn

Comments

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.

Examples

			Let S = A366413 = {A120944 \ A002110}.
This sequence is the union of the following infinite sets:
S(1)^2 * A003592 = 10^2 * A003592 = {100, 200, 400, 500, 800, 1000, ...}
                 = { m*S(1)^2 : rad(m) | S(1) }.
S(2)^2 * A003591 = 14^2 * A003591 = {196, 392, 784, 1372, 1568, ...}
                 = { m*S(2)^2 : rad(m) | S(2) }.
S(3)^2 * A003593 = 15^2 * A003593 = {225, 675, 1125, 2025, 3375, ...}
                 = { m*S(3)^2 : rad(m) | S(3) }, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A001222, A002110, A055932, A120944, A126706, A286708, A366413, A369374.

Programs

Mathematica

With[{nn = 2^14},
  Select[
    Select[
      Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Nand[EvenQ[#],
    Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

Formula

{a(n)} = { A286708 \ A055932 }.

{a(n)} = { m*s^2 : Omega(s) = omega(s) > 1, s not in A002110, rad(m) | s }.

A286708 is the union of A369374 and this sequence.