A389678 - OEIS

A389678

Powers k^m, m > 1, where k is neither squarefree nor squareful 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

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OFFSET

1,1

COMMENTS

Powers k^m, m > 1, where k is in A386294.

Intersection of A080259 and A386762 = A386762 \ A055932.

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

Contains all odd numbers in A386762.

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

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

1 400 = 20^2 2^4 * 5^2 3

2 784 = 28^2 2^4 * 7^2 3

3 1600 = 40^2 2^6 * 5^2 3

4 1936 = 44^2 2^4 * 11^2 3

5 2025 = 45^2 3^4 * 5^2 2

6 2500 = 50^2 2^2 * 5^4 3

7 2704 = 52^2 2^4 * 13^2 3

8 3136 = 56^2 2^6 * 7^2 3

9 3969 = 63^2 3^4 * 7^2 2

14 7056 = 84^2 = 2^4 * 3^2 * 7^2 5

16 8000 = 20^3 = 2^6 * 5^3 3

26 17424 = 132^2 = 2^4 * 3^2 * 11^2 5

MATHEMATICA

nn = 2^16; mm = Sqrt[nn]; i = 1; k = 2; a053669[x_] := Module[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q]; fQ[x_] := And[Length[#] > 1, a053669[x] < #[[-1, 1]], 1 == Min[#] < Max[#] &[#[[;; , -1]]]] &[FactorInteger[x]]; MapIndexed[Set[S[First[#2] ], #1] &, Select[Range@ Sqrt[nn], 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, A131605, A286708, A368089, A369417, A380456, A386294, A389394.

Sequence in context: A061042 A223481 A380887 * A390180 A037991 A200837

Adjacent sequences: A389675 A389676 A389677 * A389679 A389680 A389681

KEYWORD

nonn

AUTHOR

Michael De Vlieger, Oct 13 2025

STATUS

approved