A389341 Powers k^m, m > 1, where k is an Achilles number such that A053669(k) < A006530(k).

40000, 153664, 250000, 455625, 640000, 937024, 1265625, 1750329, 1827904, 1882384, 2458624, 4000000, 5345344, 8000000, 8340544, 9529569, 10240000, 10673289, 12446784, 14992384, 16000000, 17909824, 20820969, 25000000, 28005264, 28344976, 29246464, 30118144, 36905625

Offset

1,1

Comments

Powers k^m, m > 1, where k is in A386434.

Intersection of A080259 and A383394 = A383394 \ A055932.

Proper subset of A380456, in turn a proper subset of A369417, in turn a proper subset of A368089.

Contains all odd numbers in A383394.

Links

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

Index entries for sequences related to powerful numbers.

Example

Let q = A053669(a(n)).
Table of n, a(n) for select n:
 n       a(n)                          q
----------------------------------------
 1     40000 =  200^2 = 2^6  *  5^4    3
 2    153664 =  392^2 = 2^6  *  7^4    3
 3    250000 =  500^2 = 2^4  *  5^6    3
 4    455625 =  675^2 = 3^6  *  5^4    2
 5    640000 =  800^2 = 2^10 *  5^4    3
 6    937024 =  968^2 = 2^6  * 11^4    3
 7   1265625 = 1125^2 = 3^4  *  5^6    2
 8   1750329 = 1323^2 = 3^6  *  7^4    2
 9   1827904 = 1352^2 = 2^6  * 13^4    3
10   1882384 = 1372^2 = 2^4  *  7^6    3
11   2458624 = 1568^2 = 2^10 *  7^4    3
14   8000000 =  200^3 = 2^9  *  5^6    3

Mathematica

nn = 2^26; mm = Sqrt[nn]; i = 1; k = 2;
a053669[x_] := Module[{qx}, qx = 2; While[Divisible[x, qx], qx = NextPrime[qx]]; qx];
MapIndexed[Set[S[First[#2]], #1] &,
  Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}],
    And[GCD @@ #2[[;; , -1]] == 1,
      a053669[#1] < #2[[-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. A001597, A006530, A052486, A053669, A055932, A080259, A126706, A131605, A286708, A368089, A369417, A380456, A383394, A386434, A389226.

Sequence in context: A252332 A106772 A015328 * A251970 A384401 A270261

Adjacent sequences: A389338 A389339 A389340 * A389342 A389343 A389344

Keyword

nonn

Author

Michael De Vlieger, Oct 02 2025

Status

approved