A389312 Powers k^m, m > 1, such that k is in A303946 and A053669(k) > A006530(k).

144, 324, 576, 1728, 2304, 2916, 3600, 5184, 5832, 8100, 9216, 11664, 13824, 14400, 20736, 22500, 26244, 32400, 36864, 57600, 72900, 82944, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 186624, 202500, 216000, 230400, 236196, 248832, 291600, 331776, 360000

Offset

1,1

Comments

Intersection of A389864 and A055932 = A380446 \ A365308.

Union of disjoint subsequences A389226 and A389394.

A380446 is the union of this sequence and A365308, both disjoint.

A369374 is the union of this sequence, A365308, and A377854, all disjoint.

A363814 is the union of this sequence, A365308, A377854, and A380543, all disjoint.

Links

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

Index entries for sequences related to powerful numbers.

Example

Table of n, a(n) for n = 1..12:
 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    5184 =  72^2 = 2^6  * 3^4
 9    5832 =  18^3 = 2^3  * 3^6
10    8100 =  90^2 = 2^2  * 3^4 * 5^2
11    9216 =  96^2 = 2^10 * 3^2
12   11664 = 108^2   2^4  * 3^6

Mathematica

nn = 360000; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[#[[1, 1]] == 2, Length[#] > 1, CountDistinct[#[[;; , -1]] ] > 1, Union@ Differences@ Map[PrimePi, #[[;; , 1]] ] == {1}] &[FactorInteger[x]]; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[mm], 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, A006530, A053669, A055932, A126706, A286708, A363814, A365308, A369374, A377854, A380446, A380543, A389226, A389394.

Sequence in context: A217584 A030633 A189988 * A389394 A232892 A034285

Adjacent sequences: A389309 A389310 A389311 * A389313 A389314 A389315

Keyword

nonn

Author

Michael De Vlieger, Oct 31 2025

Status

approved