A389682 Powers k^m, m > 1, where k is neither squarefree nor squareful, and has a primorial kernel but is not a product of primorials.

324, 2916, 5832, 8100, 22500, 26244, 72900, 90000, 104976, 157464, 202500, 236196, 291600, 360000, 396900, 562500, 656100, 729000, 1102500, 1440000, 1822500, 1889568, 2125764, 2160900, 2250000, 2624400, 3375000, 3572100, 4251528, 4410000, 5062500, 5760000, 5904900

Offset

1,1

Comments

Powers k^m, m > 1, where k is in A386224.

Intersection of A056808 and A386762 = A389394 \ A389397.

Proper subset of A380452, in turn a proper subset of A369420, in turn a proper subset of A056808.

Powers k^m, m > 1, such that k is in A386762, and A053669(k) > A006530(k), but k is not a product of primorials (i.e., in A025487).

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      324 =  18^2 = 2^2 * 3^4
 2     2916 =  54^2 = 2^2 * 3^6
 3     5832 =  18^3 = 2^3 * 3^6
 4     8100 =  90^2 = 2^2 * 3^4 * 5^2
 5    22500 = 150^2 = 2^2 * 3^2 * 5^4
 6    26244 = 162^2 = 2^2 * 3^8
 7    72900 = 270^2 = 2^2 * 3^6 * 5^2
 8    90000 = 300^2 = 2^4 * 3^2 * 5^4
 9   104976 =  18^4 = 2^4 * 3^8
10   157464 =  54^3 = 2^3 * 3^9
11   202500 = 450^2 = 2^2 * 3^4 * 5^4
12   236196 = 486^2 = 2^2 * 3^10

Mathematica

nn = 6000000; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[#[[1, 1]] == 2, Length[#] > 1, Union@ Differences@ Map[PrimePi, #[[;; , 1]] ] == {1}, ReverseSort[#[[;; , -1]] ] != #[[;; , -1]], 1 == Min[#] < Max[#] &[#[[;; , -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, A025487, A053669, A055932, A056808, A126706, A131605, A286708, A369420, A380452, A386224, A389394, A389397.

Sequence in context: A250437 A064197 A380452 * A233639 A203374 A183358

Adjacent sequences: A389679 A389680 A389681 * A389683 A389684 A389685

Keyword

nonn

Author

Michael De Vlieger, Oct 17 2025

Status

approved