A389397 Powers k^m, m > 1, where k is neither squarefree nor squareful and is a product of primorials.

144, 576, 1728, 2304, 3600, 9216, 13824, 14400, 20736, 32400, 36864, 57600, 110592, 129600, 147456, 176400, 216000, 230400, 248832, 331776, 518400, 589824, 705600, 884736, 921600, 1166400, 1587600, 1728000, 2073600, 2359296, 2822400, 2985984, 3686400, 4665600

Offset

1,1

Comments

Powers k^m, m > 1, where k is in A386223.

Intersection of A025487 and A386762.

Proper subset of A368682, in turn a proper subset of A364930, in turn a proper subset of A364710.

Superset of A368508 (i.e., proper powers of superprimorials A006939).

Links

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

Index entries for sequences related to powerful numbers.

Example

 n     a(n)
-------------------------------------
 1     144 =  12^2 = 2^4  * 3^2
 2     576 =  24^2 = 2^6  * 3^2
 3    1728 =  12^3 = 2^6  * 3^3
 4    2304 =  48^2 = 2^8  * 3^2
 5    3600 =  60^2 = 2^4  * 3^2 * 5^2
 6    9216 =  96^2 = 2^10 * 3^2
 7   13824 =  24^3 = 2^9  * 3^3
 8   14400 = 120^2 = 2^6  * 3^2 * 5^2
 9   20736 =  12^4 = 2^8  * 3^4
10   32400 = 180^2 = 2^4  * 3^4 * 5^2
11   36864 = 192^2 = 2^12 * 3^2
12   57600 = 240^2 = 2^8  * 3^2 * 5^2

Mathematica

nn = 2^31; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[Length[#] > 1, 1 == Min[#] < Max[#], Times @@ MapIndexed[Prime[First[#2]]^#1 &, ReverseSort[#]] == x] &[FactorInteger[x][[;; , -1]] ]; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], fQ]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

Crossrefs

Cf. A001597, A001694, A002110, A006939, A025487, A055932, A126706, A131605, A286708, A364710, A364930, A368508, A368682, A386223, A389394.

Sequence in context: A180413 A396551 A390290 * A137885 A204398 A204391

Adjacent sequences: A389394 A389395 A389396 * A389398 A389399 A389400

Keyword

nonn

Author

Michael De Vlieger, Oct 10 2025

Status

approved