A393201 Numbers k that are neither squarefree nor prime powers, such that k/rad(k) != lpf(k), where rad = A007947 and lpf = A020639.

18, 24, 36, 40, 48, 50, 54, 56, 72, 75, 80, 88, 90, 96, 98, 100, 104, 108, 112, 120, 126, 135, 136, 144, 147, 150, 152, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 208, 216, 224, 225, 232, 234, 240, 242, 245, 248, 250, 252, 264, 270, 272, 280, 288, 294

Offset

1,1

Comments

These numbers k are such that there exists at least 1 number j < k such that rad(j) = rad(k) but j does not divide k. This number j is in A366825.

This is to say that k is such that A008479(k) > 2, however, there are prime powers p^m, m > 2 (p^m in A246549) that also have the same property.

The union of this sequence and A246549 comprises the sequence of numbers k such that A008479(k) > 2.

This sequence is {k : 1 < omega(k) < bigomega(k), k/rad(k) != lpf(k)}, where omega = A001221 and bigomega = A001222.

This sequence is the union of sequences {k*m : 1 < omega(k) = bigomega(k), rad(m) | k, m > lpf(k)} (for k in A120944).

Numbers k in this sequence appear after rad(k) and rad(k)*lpf(k) in the list of numbers that have the same squarefree kernel as k. For example, 18 (24, 36, 48, etc.) appears after 6 and 12 in A033845, while 40 (50, 80, 100, etc.) appears after 10 and 20 in A033846.

The ordered prime signature of a(n) (i.e., row a(n) of A124010) is not of the form {2,1,...,1}, i.e., 2 followed by at least one 1.

Local complement of A366825 with respect to A126706.

Numbers k in A126706 such that A003557(k) != A020639(k).

Superset of A286708 (powerful numbers that are not prime powers).

Links

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

Example

Let c(n) = A008479(n), the length of row n of A369609.
Table of n, a(n), c(a(n)) for n = 1..12:
   n  a(n) c(a(n))  Row a(n) of A369609
  ----------------------------------------------
   1   18      3    {6, 12, 18}
   2   24      4    {6, 12, 18, 24}
   3   36      5    {6, 12, 18, 24, 36}
   4   40      3    {10, 20, 40}
   5   48      6    {6, 12, 18, 24, 36, 48}
   6   50      4    {10, 20, 40, 50}
   7   54      7    {6, 12, 18, 24, 36, 48, 54}
   8   56      3    {14, 28, 56}
   9   72      8    {6, 12, 18, 24, 36, 48, 54, 72}
  10   75      3    {15, 45, 75}
  11   80      5    {10, 20, 40, 50, 80}
  12   88      3    {22, 44, 88}

Mathematica

s = Select[Range[300], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; Select[s, Nand[First[#] == 2, AllTrue[Rest[#], # == 1 &]] &@ FactorInteger[#][[All, -1]] &]

Crossrefs

Cf. A001221, A001222, A003557, A007947, A008479, A013929, A020639, A024619, A120944, A124010, A126706, A246549, A286708, A366825, A369609.

Sequence in context: A334015 A162331 A092536 * A214620 A360768 A359409

Adjacent sequences: A393198 A393199 A393200 * A393202 A393203 A393204

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 17 2026

Status

approved