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

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

Author

Michael De Vlieger, Jan 28 2024

Keywords

• nonn

Comments

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

Superset of A369417.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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[#]} &] ]

Formula

{a(n)} = A286708 \ A025487.