A387901 Powerful numbers with a squarefree kernel that is a product of the smallest primes.

1, 4, 8, 16, 32, 36, 64, 72, 108, 128, 144, 216, 256, 288, 324, 432, 512, 576, 648, 864, 900, 972, 1024, 1152, 1296, 1728, 1800, 1944, 2048, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4096, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8192, 8748, 9000

Offset

1,2

Comments

Intersection of A001694 and A055932.

Union of A000079 and A369374 without the number 2.

Superset of A181800.

The empty product appears since 1 is the product of no primes at all and 1^2*1^3 = 1.

Smallest term with k distinct prime factors is A061742(k).

Links

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

Michael De Vlieger, Efficient Mathematica algorithm.

Example

Table of n, a(n) for select n, with p-adic valuation for p | a(n):
             p-adic valuation
 n     a(n)  2 3 5 7
-------------------
 1       1
 2       4   2
 3       8   3
 4      16   4
 5      32   5
 6      36   2.2
 8      72   3.2
 9     108   2.3
11     144   4.2
21     900   2.2.2
27    1800   3.2.2
32    2700   2.3.2
35    3600   4.2.2
81   44100   2.2.2.2

Mathematica

With[{nn = 9000},
  Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Or[IntegerQ@ Log2[#],
      And[EvenQ[#], Union@ Differences@
        PrimePi[FactorInteger[#][[All, 1]] ] == {1}] ] &] ]

Crossrefs

Cf. A000079, A001694, A055932, A061742, A181800, A369374.

Sequence in context: A359777 A391627 A390010 * A181800 A319180 A075090

Adjacent sequences: A387898 A387899 A387900 * A387902 A387903 A387904

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 11 2025

Status

approved