A391756 - OEIS

%I #9 Dec 20 2025 10:39:00

%S 36,100,144,196,216,324,400,484,576,676,784,900,1000,1156,1296,1444,

%T 1600,1728,1764,1936,2116,2304,2500,2704,2744,2916,3136,3364,3600,

%U 3844,4356,4624,4900,5184,5476,5776,5832,6084,6400,6724,7056,7396,7744,7776,8000,8100

%N Powers k^m, m > 1, of even k that are not perfect powers.

%C Powers k^m, m > 1, of even k in A007916.

%C Even terms in A131605.

%C Disjoint union of A390956, A391320, and A391376.

%C A075090 is the union of this sequence and A000079.

%C A390010 is the union of this sequence and A390952.

%C A131605 is the union of this sequence and A216419 (odd terms in A131605).

%H Michael De Vlieger, <a href="/A391756/b391756.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%e Table of n, a(n) for n = 1..12:

%e  n   a(n)

%e ----------------------------------

%e  1    36 =  6^2 = 2^2 *  3^2

%e  2   100 = 10^2 = 2^2 *  5^2

%e  3   144 = 12^2 = 2^4 *  3^2

%e  4   196 = 14^2 = 2^2 *  7^2

%e  5   216 =  6^3 = 2^3 *  3^3

%e  6   324 = 18^2 = 2^2 *  3^4

%e  7   400 = 20^2 = 2^4 *  5^2

%e  8   484 = 22^2 = 2^2 * 11^2

%e  9   576 = 24^2 = 2^6 *  3^2

%e 10   676 = 26^2 = 2^2 * 13^2

%e 11   784 = 28^2 = 2^4 *  7^2

%e 12   900 = 30^2 = 2^2 *  3^2 * 5^2

%t With[{nn = 10000}, Union@ Flatten@ Table[If[And[EvenQ[#1], Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]}, #1, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]

%Y Cf. A000079, A001597, A001694, A007916, A075090, A126706, A131605, A216419, A286708, A390010, A390952, A390956, A391320, A391376.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Dec 18 2025