A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

72, 108, 144, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1000, 1125, 1152, 1296, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2744, 2888, 2916, 3087, 3136, 3200, 3267, 3375, 3456

Offset

1,1

Comments

A001694 \ A246547 = A286708, i.e., A286708 contains powerful numbers without perfect prime powers. Hence, this sequence is a proper subset of A286708 which in turn is contained in A126706.

Numbers k in A286708 are such that rad(k)^2 | k. Numbers in this sequence are such that k != A120944(m)^2 for some m, where A120944 is the sequence of squarefree composites.

Links

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

Formula

A286708 = union of A177492 and this sequence.

A001694 = union of A246547, A177492, and this sequence.

A126706 = union of A332785, A177492, and this sequence.

Example

The number 36 is not in the sequence since 36/rad(36) = 36/6 = 6, squarefree.
a(1) = 72 since 72/rad(72) = 72/6 = 12 is nonsquarefree.
a(2) = 108 since 108/rad(108) = 108/6 = 18 is nonsquarefree.
a(4) = 200 since 200/rad(200) = 200/10 = 20 is nonsquarefree, etc.

Mathematica

With[{nn = 3300},
  Select[
    Select[Rest@ Union@ Flatten@
      Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Not@ SquareFreeQ[#/(Times @@ FactorInteger[#][[;; , 1]])] &] ]

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]);
isok(k) = ispowerful(k) && !isprimepower(k) && !issquarefree(k/rad(k)); \\ Michel Marcus, Jun 05 2024

Crossrefs

Cf. A001694, A007947, A013929, A120944, A126708, A177492, A126708, A246547, A286708, A332785.

Sequence in context: A357460 A216426 A308053 * A356871 A359280 A307758

Adjacent sequences: A372401 A372402 A372403 * A372405 A372406 A372407

Keyword

nonn,easy

Author

Michael De Vlieger, Jun 04 2024

Status

approved