A369374 Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.

36, 72, 108, 144, 216, 288, 324, 432, 576, 648, 864, 900, 972, 1152, 1296, 1728, 1800, 1944, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8748, 9000, 9216, 10368, 10800, 11664, 13500, 13824, 14400, 15552, 16200

Offset: 1

Author

Michael De Vlieger, Jan 22 2024

Keywords

• nonn

Comments

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).

Union of the product of the squares of primorials P(n)^2, n > 1, and the set of prime(n)-smooth numbers.

Superset of A364930.

Proper subset of A367268, which in turn is a proper subset of A126706.

Examples

			This sequence is the union of the following infinite sets:
P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
                 = { m*P(2)^2 : rad(m) | P(2) }.
P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
                 = { m*P(3)^2 : rad(m) | P(3) }.
P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
                 = { m*P(4)^2 : rad(m) | P(4) }, etc.
		

Links

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

Crossrefs

Cf. A001221, A001222, A001694, A002110, A007947, A055932, A126706, A286708, A364930, A367268, A369417.

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],
    And[EvenQ[#],
      Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

Formula

{a(n)} = { m*P(n)^2 : P(n) = Product_{j = 1..n} prime(n), rad(m) | P(n), n > 1 }.

Intersection of A286708 and A055932.

A286708 is the union of A369417 and this sequence.