A367268 -id:A367268 - cp's OEIS frontend

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

Michael De Vlieger, Jan 22 2024

nonn

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.

			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.

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

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

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}] &] ]

{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.

A368089 Numbers k neither squarefree nor prime powers whose squarefree kernel is not a primorial.

Original entry on oeis.org

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 84, 88, 92, 98, 99, 100, 104, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212, 220, 224, 225, 228, 232

Offset: 1

Michael De Vlieger, Jan 20 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1 such that squarefree kernel rad(k) is not in A002110, where Omega = A001222, omega = A001221, and rad(k) = A007947(k).

Contains numbers k in A126706 that are not of the form m*P(n), where P(n) = A002110(n), rad(m) | P(n), m > 1, n > 1.

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

Cf. A001221, A001222, A002110, A007947, A055932, A126706, A367268.

Select[Select[Range[240], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], Nand[EvenQ[#], Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &]

This sequence is { A126706 \ A055932 } = { A126706 \ A367268 }.

A369636 Powerful numbers k that are neither prime powers nor products of primorials.

Original entry on oeis.org

100, 108, 196, 200, 225, 324, 392, 400, 441, 484, 500, 648, 675, 676, 784, 800, 968, 972, 1000, 1089, 1125, 1156, 1225, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1764, 1936, 1944, 2000, 2025, 2116, 2312, 2500, 2601, 2700, 2704, 2744, 2888, 2916, 3025, 3087, 3136

Offset: 1

Michael De Vlieger, Jan 28 2024

nonn

Proper subset of A367268 which is in turn a proper subset of A080259.

Superset of A369417.

			Let P(n) = A002110(n).
36 = 6^2 = P(2)^2 is a product of primorials and not in the sequence.
72 = 2 * 6^2 = P(1) * P(2)^2 is a product of primorials and not in the sequence.
a(1) = 100 = 2^2 * 5^2 is not a product of primorials.
a(2) = 108 = 2^2 * 3^3 is not a product of primorials, etc.

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

Cf. A001694, A002110, A025487, A080259, A126706, A286708, A367268, A369417.

With[{nn = 3200}, Select[
 Select[
   Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
   Not@*PrimePowerQ],
 Nand[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
   AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
   {#, FactorInteger[#]} &] ]

{a(n)} = A286708 \ A025487.

cp's OEIS Frontend

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

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A368089 Numbers k neither squarefree nor prime powers whose squarefree kernel is not a primorial.

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A369636 Powerful numbers k that are neither prime powers nor products of primorials.

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Keywords

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Programs

Mathematica

Formula