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%I #9 Jan 25 2026 20:39:18
%S 288,432,648,800,864,972,1152,1568,1944,2000,3200,3456,3872,4000,4608,
%T 5000,5408,5488,6075,6272,6912,7200,8748,9248,10125,10800,10976,11552,
%U 11907,12500,12800,14112,15488,16000,16200,16875,16928,17496,18000,18432,19208,21168
%N Achilles numbers that are neither 4-free nor 4-full.
%C Intersection of A052486 (Achilles numbers) and A391115 (numbers that are neither 4-free nor 4-full).
%C A052486 is the union of this sequence, A375073 (noncubefull Achilles numbers), and A391011 (4-full Achilles numbers).
%C This sequence intersects both A388293 (cubefull Achilles numbers) and A390539 (noncubefull Achilles numbers).
%C Numbers whose set of prime power factor exponents m is setwise coprime, contains at least one m >= 4, but not all m >= 4, yet all m > 1.
%H Michael De Vlieger, <a href="/A392134/b392134.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 select n:
%e n a(n)
%e ---------------------------
%e 1 288 = 2^5 * 3^2
%e 2 432 = 2^4 * 3^3
%e 3 648 = 2^3 * 3^4
%e 4 800 = 2^5 * 5^2
%e 5 864 = 2^5 * 3^3
%e 6 972 = 2^2 * 3^5
%e 7 1152 = 2^7 * 3^2
%e 8 1568 = 2^5 * 7^2
%e 9 1944 = 2^3 * 3^5
%e 10 2000 = 2^4 * 5^3
%e 19 6075 = 3^5 * 5^2
%e 22 7200 = 2^5 * 3^2 * 5^2
%t With[{nn = 25000}, Union@ Flatten@ Table[If[And[GCD @@ # == 1, 0 < Count[#, _?(# > 3 &)] < Length[#]] &[FactorInteger[#][[;; , -1]]], #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]} ] ]
%o (PARI) is_A392134(n) = if(n<=1, 0, (e->(1==gcd(e) && 3==bitor(vecmin(e),1) && vecmax(e)>3))(factor(n)[,2])); \\ _Antti Karttunen_, Jan 22 2026
%Y Cf. A375073, A388293, A390539, A391011.
%Y Supersets: A001694, A013929, A024619, A046101, A052486, A126706, A286708, A391115.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Jan 15 2026