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%I #9 Mar 15 2026 21:50:23
%S 1800,2700,3528,4500,5292,5400,7200,8712,9000,9800,10584,10800,12168,
%T 12348,13068,13500,14112,16200,18000,18252,20808,21168,24200,24300,
%U 24500,24696,25992,26136,28800,31212,31752,33075,33800,34300,34848,36504,37044,38088,38988
%N Achilles numbers whose prime power factor exponents are not all pairwise coprime.
%C Proper subset of A393708 = A052486 intersect A000977, Achilles numbers with at least 3 distinct prime factors.
%C Smallest a(n) with m distinct prime factors is 2*A002110(m)^2 = 2*A061742(m) for m > 2.
%H Michael De Vlieger, <a href="/A394000/b394000.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>.
%F {a(n)} = A052486 \ A393999.
%e Let s = A052486.
%e Table of n, a(n) for select n:
%e n a(n)
%e --------------------------------------------
%e 1 s(20) = 1800 = 2^3 * 3^2 * 5^2
%e 2 s(25) = 2700 = 2^2 * 3^3 * 5^2
%e 3 s(31) = 3528 = 2^3 * 3^2 * 7^2
%e 4 s(36) = 4500 = 2^2 * 3^2 * 5^3
%e 5 s(40) = 5292 = 2^2 * 3^3 * 7^2
%e 6 s(42) = 5400 = 2^3 * 3^3 * 5^2
%e 7 s(50) = 7200 = 2^5 * 3^2 * 5^2
%e 8 s(54) = 8712 = 2^3 * 3^2 * 11^2
%e 9 s(57) = 9000 = 2^3 * 3^2 * 5^3
%e 12 s(64) = 10800 = 2^4 * 3^3 * 5^2
%e 72 s(226) = 81675 = 3^3 * 5^2 * 11^2
%e 76 s(237) = 88200 = 2^3 * 3^2 * 5^2 * 7^2
%t nn = 2^16; fQ[x_] := And[GCD @@ # == 1, ! CoprimeQ @@ #] &@FactorInteger[x][[All, -1]]; Union@ Flatten@ Table[If[fQ[#], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ]
%Y Cf. A000977, A001694, A052486, A126706, A286708, A393708, A393999.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Mar 06 2026