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

1,1

Comments

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

Superset of A369417.

Links

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

Formula

{a(n)} = A286708 \ A025487.

Example

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.

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[#]} &] ]

Crossrefs

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

Sequence in context: A204586 A204587 A204588 * A352440 A204589 A134496

Adjacent sequences: A369633 A369634 A369635 * A369637 A369638 A369639

Keyword

nonn

Author

Michael De Vlieger, Jan 28 2024

Status

approved