A389394 Powers k^m, m > 1, where k is neither squarefree nor squareful and A006530(k) < A053669(k).

144, 324, 576, 1728, 2304, 2916, 3600, 5832, 8100, 9216, 13824, 14400, 20736, 22500, 26244, 32400, 36864, 57600, 72900, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 202500, 216000, 230400, 236196, 248832, 291600, 331776, 360000, 396900, 518400, 562500

Offset

1,1

Comments

Powers k^m, m > 1, where k is in A380543.

Intersection of A055932 and A386762.

Superset of A389397, thus also superset of A368508 (i.e., proper powers of superprimorials A006939).

Proper subset of A380446, in turn a proper subset of A369374, in turn a proper subset of A363814.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..18030

Index entries for sequences related to powerful numbers.

Example

 n     a(n)
------------------------------------
 1     144 =  12^2 = 2^4  * 3^2
 2     324 =  18^2 = 2^2  * 3^4
 3     576 =  24^2 = 2^6  * 3^2
 4    1728 =  12^3 = 2^6  * 3^3
 5    2304 =  48^2 = 2^8  * 3^2
 6    2916 =  54^2 = 2^2  * 3^6
 7    3600 =  60^2 = 2^4  * 3^2 * 5^2
 8    5832 =  18^3 = 2^3  * 3^6
 9    8100 =  90^2 = 2^2  * 3^4 * 5^2
10    9216 =  96^2 = 2^10 * 3^2
11   13824 =  24^3 = 2^9  * 3^3
12   14400 = 120^2 = 2^6  * 3^2 * 5^2

Mathematica

nn = 2^20; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[#[[1, 1]] == 2, Length[#] > 1, Union@ Differences@ Map[PrimePi, #[[;; , 1]] ] == {1}, 1 == Min[#] < Max[#] &[#[[;; , -1]] ] ] &[FactorInteger[x] ]; 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, A006530, A006939, A053669, A055932, A126706, A131605, A286708, A363814, A368508, A369374, A380446, A380543, A386762, A389397.

Sequence in context: A030633 A189988 A389312 * A232892 A034285 A211469

Adjacent sequences: A389391 A389392 A389393 * A389395 A389396 A389397

Keyword

nonn

Author

Michael De Vlieger, Oct 10 2025

Status

approved