A394482 Achilles numbers whose prime power factor multiplicities are not pairwise indivisible.

10800, 16200, 18000, 21168, 31752, 40500, 43200, 45000, 49392, 52272, 64800, 67500, 72000, 73008, 78408, 84672, 97200, 98000, 109512, 111132, 124848, 127008, 137200, 145800, 155952, 172800, 172872, 180000, 187272, 190512, 191664, 197568, 209088, 228528, 233928

Offset

1,1

Comments

Proper subset of A394000, in turn a proper subset of A393708.

Smallest a(n) with m distinct prime factors is 12*A002110(m) for m > 2.

Intersection of A052486 and A317101 = A052486 \ A317616.

Links

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

Index entries for sequences related to powerful numbers.

Example

Let s = A052486.
Table of n, a(n) for select n:
   n                          a(n)
  ----------------------------------------------------------------------
      1       s(64) =       10800 = 2^4 * 3^3 * 5^2
      2       s(84) =       16200 = 2^3 * 3^4 * 5^2
      3       s(89) =       18000 = 2^4 * 3^2 * 5^3
      4       s(98) =       21168 = 2^4 * 3^3 * 7^2
      5      s(127) =       31752 = 2^3 * 3^4 * 7^2
     59      s(625) =      496125 = 3^4 * 5^3 * 7^2
     62      s(643) =      529200 = 2^4 * 3^3 * 5^2 * 7^2
   1555     s(8363) =    64033200 = 2^4 * 3^3 * 5^2 * 7^2 * 11^2
  27221   s(116468) = 10821610800 = 2^4 * 3^3 * 5^2 * 7^2 * 11^2 * 13^2

Mathematica

nn = 2^18; fQ[x_] := And[GCD @@ # == 1, Count[Tuples[#, 2], _?(And[UnsameQ @@ #, Divisible @@ #] &)] > 0] &[FactorInteger[x][[All, -1]] ]; Union@ Flatten@ Table[If[fQ[#], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ]

Crossrefs

Cf. A001694, A013929, A024619, A052486, A126706, A286708, A317101, A393708, A394000.

Sequence in context: A010089 A216142 A340303 * A093770 A190115 A203394

Adjacent sequences: A394479 A394480 A394481 * A394483 A394484 A394485

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 24 2026

Status

approved