A390180 Powers k^m, m > 1, such that k is in A303946 and A053669(k) < A006530(k).

400, 784, 1600, 1936, 2025, 2500, 2704, 3136, 3969, 4624, 5625, 5776, 6400, 7056, 7744, 8000, 8464, 9604, 9801, 10816, 12544, 13456, 13689, 15376, 15876, 17424, 18225, 18496, 19600, 21609, 21904, 21952, 23104, 23409, 24336, 25600, 26896, 28224, 29241, 29584, 30625, 30976, 33856, 35344, 35721, 39204, 40000

Offset

1,1

Comments

Distinct from A389678; a(47) = 40000 = A389341(1), while A389678(47) = 41616.

Intersection of A389864 and A080259 = A389864 \ A055932 = A389864 \ A389312 = A380456 \ A365745.

Union of disjoint subsequences A389341 and A389678.

This sequence is A389864 \ A389312.

A380456 is the union of this sequence and A365745, both disjoint.

A369417 is the union of this sequence, A365745, and A386434, all disjoint.

A368089 is the union of this sequence, A365745, A386434, and A386294, 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 select n:
 n     a(n)
-----------------------------------
 1     400 =  20^2 = 2^4 * 5^2
 2     784 =  28^2 = 2^4 * 7^2
 3    1600 =  40^2 = 2^6 * 5^2
 4    1936 =  44^2 = 2^4 * 11^2
 5    2025 =  45^2 = 3^4 * 5^2
 6    2500 =  50^2 = 2^2 * 5^4
 7    2704 =  52^2 = 2^4 * 13^2
 8    3136 =  56^2 = 2^6 * 7^2
 9    3969 =  63^2 = 3^4 * 7^2
11    5625 =  75^2 = 3^2 * 5^4
14    7056 =  84^2 = 2^4 * 3^2 * 7^2
47   40000 = 200^2 = 2^6 * 5^4

Mathematica

a053669[x_] := Block[{qx}, qx = 2; While[Divisible[x, qx], qx = NextPrime[qx]]; qx]; nn = 2^15; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[Length[#] > 1, CountDistinct[#[[;; , -1]] ] > 1, a053669[x] < #[[-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, A080259, A126706, A286708, A365745, A368089, A380456, A386294, A386434, A389312, A389341, A389678, A389864.

Sequence in context: A223481 A380887 A389678 * A037991 A200837 A250847

Adjacent sequences: A390177 A390178 A390179 * A390181 A390182 A390183

Keyword

nonn

Author

Michael De Vlieger, Nov 03 2025

Status

approved