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A046101 Biquadrateful numbers.

%I A046101 #52 Feb 16 2025 08:32:38
%S A046101 16,32,48,64,80,81,96,112,128,144,160,162,176,192,208,224,240,243,256,
%T A046101 272,288,304,320,324,336,352,368,384,400,405,416,432,448,464,480,486,
%U A046101 496,512,528,544,560,567,576,592,608,624,625,640,648,656,672,688,704
%N A046101 Biquadrateful numbers.
%C A046101 The convention in the OEIS is that squareful, cubeful (A046099), biquadrateful, ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full (A036966), 4-full (A036967) are used for Golomb's notion of powerful numbers (A001694, see references there), when each prime factor occurs to a power > 1. - _M. F. Hasler_, Feb 12 2008
%C A046101 Also solutions to equation tau_{-3}(n)=0, where tau_{-3} is A007428. - _Enrique Pérez Herrero_, Jan 19 2013
%C A046101 Sum_{n>0} 1/a(n)^s = Zeta(s) - Zeta(s)/Zeta(4s). - _Enrique Pérez Herrero_, Jan 21 2013
%C A046101 A051903(a(n)) > 3. - _Reinhard Zumkeller_, Sep 03 2015
%C A046101 The asymptotic density of this sequence is 1 - 1/zeta(4) = 1 - 90/Pi^4 = 0.076061... - _Amiram Eldar_, Jul 09 2020
%H A046101 T. D. Noe, <a href="/A046101/b046101.txt">Table of n, a(n) for n = 1..1000</a>
%H A046101 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Biquadratefree.html">Biquadratefree.</a>
%p A046101 with(NumberTheory):
%p A046101 isBiquadrateful := n -> is(denom(Radical(n) / LargestNthPower(n, 2)) <> 1):
%p A046101 select(isBiquadrateful, [`$`(1..704)]);  # _Peter Luschny_, Jul 12 2022
%t A046101 lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>3, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 15 2008 *)
%t A046101 Select[Range[1000],Max[Transpose[FactorInteger[#]][[2]]]>3&] (* _Harvey P. Dale_, May 25 2014 *)
%o A046101 (Haskell)
%o A046101 a046101 n = a046101_list !! (n-1)
%o A046101 a046101_list = filter ((> 3) . a051903) [1..]
%o A046101 -- _Reinhard Zumkeller_, Sep 03 2015
%o A046101 (PARI) is(n)=n>9 && vecmax(factor(n)[,2])>3 \\ _Charles R Greathouse IV_, Sep 03 2015
%o A046101 (Python)
%o A046101 from sympy import mobius, integer_nthroot
%o A046101 def A046101(n):
%o A046101     def f(x): return n+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))
%o A046101     m, k = n, f(n)
%o A046101     while m != k:
%o A046101         m, k = k, f(k)
%o A046101     return m # _Chai Wah Wu_, Aug 05 2024
%Y A046101 Cf. A046100, A046099, A036967, A001694, A051903, A215267.
%K A046101 nonn
%O A046101 1,1
%A A046101 _Eric W. Weisstein_