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A355265 Bicubeful numbers.

%I A355265 #20 Jul 13 2022 09:10:27
%S A355265 64,128,192,256,320,384,448,512,576,640,704,729,768,832,896,960,1024,
%T A355265 1088,1152,1216,1280,1344,1408,1458,1472,1536,1600,1664,1728,1792,
%U A355265 1856,1920,1984,2048,2112,2176,2187,2240,2304,2368,2432,2496,2560,2624,2688,2752
%N A355265 Bicubeful numbers.
%C A355265 Let lp(n, e) denote the largest positive integer b such that b^e divides n. For example for e = 1, 2, 3, 4 the sequences (lp(n, e), n >= 1) are A000027, A000188, A053150, and A053164. Let rad(n) = A007947(n) be the squarefree kernel of n. k is in this sequence if lp(n, 3) does not divide rad(n). The case e = 1 gives A013929, and the case e = 2 is A046101.
%C A355265 The asymptotic density of this sequence is 1 - 1/zeta(6) = 1 - 945/Pi^6 = 0.017047... . - _Amiram Eldar_, Jul 13 2022
%H A355265 Amiram Eldar, <a href="/A355265/b355265.txt">Table of n, a(n) for n = 1..10000</a>
%F A355265 A number k is bicubeful iff it is divisible by the 6th power of an integer > 1.
%e A355265 n = 512 = 2^9, rad(n) = 2, lp(n, 3) = 8 since n/8^3 = 1. But 8 does not divide 2.
%e A355265 n = 704 = 2^6*11, rad(n) = 22, lp(n, 3) = 4 since n/4^3 = 11. But 4 does not divide 22.
%p A355265 with(NumberTheory):
%p A355265 isBicubeful := n -> irem(Radical(n), LargestNthPower(n, 3)) <> 0:
%p A355265 select(isBicubeful, [`$`(1..2752)]);
%t A355265 bicubQ[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # > 5 &]; Select[Range[3000], bicubQ] (* _Amiram Eldar_, Jul 13 2022 *)
%o A355265 (Python)
%o A355265 from itertools import count, islice
%o A355265 from sympy import factorint
%o A355265 def A355265_gen(startvalue=1): # generator of terms >= startvalue
%o A355265     return filter(lambda n:any(map(lambda m:m>5,factorint(n).values())),count(max(startvalue,1)))
%o A355265 A355265_list = list(islice(A355265_gen(),30)) # _Chai Wah Wu_, Jul 12 2022
%Y A355265 Cf. A007947, A000188, A053150, A053164, A013929, A046101 (biquadrateful).
%Y A355265 Cf. A013664, A343359.
%K A355265 nonn
%O A355265 1,1
%A A355265 _Peter Luschny_, Jul 12 2022