A389260 Powers k^m, m > 1, where k is an Achilles number that is a product of primorials.

5184, 82944, 186624, 373248, 746496, 1327104, 3240000, 6718464, 11943936, 21233664, 23887872, 26873856, 29160000, 47775744, 51840000, 80621568, 107495424, 116640000, 241864704, 339738624, 466560000, 644972544, 764411904, 829440000, 967458816, 1528823808, 1719926784

Offset

1,1

Comments

Powers k^m, with k in A378002 and m > 1.

Intersection of A025487 and A383394.

Proper subset of A389260, since A025487 is a proper subset of A055932.

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

Numbers j such that omega(j) > 1, whose prime power factor exponents have a greatest common factor g > 1, but m^(1/g) has prime power factor exponents exceed 1 and are coprime.

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 <= 12:
 n        a(n)
-----------------------------------------
 1       5184 =   72^2 = 2^6  * 3^4
 2      82944 =  288^2 = 2^10 * 3^4
 3     186624 =  432^2 = 2^8  * 3^6
 4     373248 =   72^3 = 2^9  * 3^6
 5     746496 =  864^2 = 2^10 * 3^6
 6    1327104 = 1152^2 = 2^14 * 3^4
 7    3240000 = 1800^2 = 2^6  * 3^4 * 5^4
 8    6718464 = 2592^2 = 2^10 * 3^8
 9   11943936 = 3456^2 = 2^14 * 3^6
10   21233664 = 4608^2 = 2^18 * 3^4
11   23887872 =  288^3 = 2^15 * 3^6
12   26873856 =   72^4 = 2^12 * 3^8

Mathematica

nn = 2^31; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[Length[#] > 1, GCD @@ # == 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, ReverseSort[#]] == x] &[FactorInteger[x][[;; , -1]] ]; MapIndexed[Set[S[First[#2]], #1] &, Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], 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, A025487, A052486, A055932, A126706, A131605, A286708, A364710, A364930, A368682, A369420, A378002, A383394, A389260.

Sequence in context: A175745 A190464 A389226 * A031570 A066167 A392519

Adjacent sequences: A389257 A389258 A389259 * A389261 A389262 A389263

Keyword

nonn

Author

Michael De Vlieger, Oct 01 2025

Status

approved