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A046101
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Biquadrateful numbers.
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25
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16, 32, 48, 64, 80, 81, 96, 112, 128, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 640, 648, 656, 672, 688, 704
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
The asymptotic density of this sequence is 1 - 1/zeta(4) = 1 - 90/Pi^4 = 0.076061... - Amiram Eldar, Jul 09 2020
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LINKS
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MAPLE
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with(NumberTheory):
isBiquadrateful := n -> is(denom(Radical(n) / LargestNthPower(n, 2)) <> 1):
select(isBiquadrateful, [`$`(1..704)]); # Peter Luschny, Jul 12 2022
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MATHEMATICA
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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 *)
Select[Range[1000], Max[Transpose[FactorInteger[#]][[2]]]>3&] (* Harvey P. Dale, May 25 2014 *)
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PROG
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(Haskell)
a046101 n = a046101_list !! (n-1)
a046101_list = filter ((> 3) . a051903) [1..]
(Python)
from sympy import mobius, integer_nthroot
def f(x): return n+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x, 4)[0]+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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