A389226 - OEIS

A389226

Powers k^m, m > 1, where k is an Achilles number whose squarefree kernel is a primorial.

5184, 11664, 82944, 186624, 373248, 419904, 746496, 944784, 1259712, 1327104, 3240000, 3779136, 6718464, 7290000, 11943936, 15116544, 20250000, 21233664, 23887872, 26873856, 29160000, 47775744, 51840000, 76527504, 80621568, 81000000, 107495424, 116640000, 136048896

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Powers k^m, with k in A377854 and m > 1, that is where k is an Achilles number whose squarefree kernel rad(k) is a primorial (i.e., in A002110), with rad = A007947.

Powers k^m, with m > 1 and k an Achilles number such that A053669(k) > gpf(k), with gpf = A006530.

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..10000

Index entries for sequences related to powerful numbers.

EXAMPLE

n a(n)

----------------------------------

1 5184 = 72^2 = 2^6 * 3^4

2 11664 = 108^2 = 2^4 * 3^6

3 82944 = 288^2 = 2^10 * 3^4

4 186624 = 432^2 = 2^8 * 3^6

5 373248 = 72^3 = 2^9 * 3^6

6 419904 = 648^2 = 2^6 * 3^8

7 746496 = 864^2 = 2^10 * 3^6

8 944784 = 972^2 = 2^4 * 3^10

9 1259712 = 108^3 = 2^6 * 3^9

10 1327104 = 1152^2 = 2^14 * 3^4

11 3240000 = 1800^2 = 2^6 * 3^4 * 5^4

12 3779136 = 1944^2 = 2^6 * 3^10

MATHEMATICA

nn = 2^30; mm = Sqrt[nn]; i = 1; k = 2;

MapIndexed[Set[S[First[#2]], #1] &,

Select[

Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}],

And[#[[1, 1]] == 2, Length[#] > 1,

Union@ Differences@ Map[PrimePi, #[[;; , 1]] ] == {1},

Apply[GCD, #[[;; , -1]] ] == 1] &[FactorInteger[#]] &] ];

Union@ Reap[

While[j = 2;

While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2,

k++; i++] ][[-1, 1]]

CROSSREFS

Cf. A002110, A006530, A007947, A052486, A053669, A055932, A126706, A131605, A286708, A363384, A369374, A377854, A380446, A383394.

Sequence in context: A390435 A175745 A190464 * A389260 A031570 A066167

Adjacent sequences: A389223 A389224 A389225 * A389227 A389228 A389229

KEYWORD

nonn

AUTHOR

Michael De Vlieger, Sep 30 2025

STATUS

approved