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%I #9 Feb 04 2026 13:39:54
%S 288,800,972,1152,1568,3200,3872,4608,5408,6075,6272,7200,8748,9248,
%T 10800,11552,11907,12500,12800,14112,15488,16200,16928,18000,18432,
%U 21168,21600,21632,24300,25088,26912,28125,28800,29403,30752,31752,34848,36000,36992,39200
%N Achilles numbers that are neither biquadratefree nor cubefull.
%C Intersection of A390539 (noncubefull Achilles numbers) and A392134 (Achilles numbers that are neither 4-free nor 4-full).
%C Intersection of A362147 (noncubefull numbers), A052486 (Achilles numbers), and A391115 (numbers that are neither 4-free nor 4-full).
%C Let omega = A001221 be the number of distinct prime factors p | k. Let p^m | k but p^(m+1) not divide k and define Q to be the multiset of m for all p | k. Define c(Q) to be the number of m such that m >= 4. This sequence is {k : gcd(Q) = 1, min(Q) = 2, 0 < c(Q) < omega(k)}.
%H Michael De Vlieger, <a href="/A392907/b392907.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 n, a(n), n = 1..12:
%e n a(n)
%e ----------------------------
%e 1 288 = 2^5 * 3^2
%e 2 800 = 2^5 * 5^2
%e 3 972 = 2^2 * 3^5
%e 4 1152 = 2^7 * 3^2
%e 5 1568 = 2^5 * 7^2
%e 6 3200 = 2^7 * 5^2
%e 7 3872 = 2^5 * 11^2
%e 8 4608 = 2^9 * 3^2
%e 9 5408 = 2^5 * 13^2
%e 10 6075 = 3^5 * 5^2
%e 11 6272 = 2^7 * 7^2
%e 12 7200 = 2^5 * 3^2 * 5^2
%t nn = 40000; Union@ Flatten@ Table[If[And[Min[#2] == 2, 0 < Count[#2, _?(# > 3 &)] < Length[#2], GCD @@ #2 == 1], #1, Nothing] & @@ {#, FactorInteger[#][[;; , -1]]} &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]
%o (PARI) is_A392907(n) = if(n<=1, 0, (e->(1==gcd(e) && vecmin(e)==2 && vecmax(e)>3))(factor(n)[,2])); \\ _Antti Karttunen_, Jan 26 2026
%Y Cf. A001694, A013929, A024619, A052486, A126706, A286708, A362147, A390539, A391115, A392134, A392908.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, Jan 26 2026