OFFSET
1,1
COMMENTS
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
Also solutions to equation tau_{-3}(n)=0, where tau_{-3} is A007428. - Enrique Pérez Herrero, Jan 19 2013
Sum_{n>0} 1/a(n)^s = Zeta(s) - Zeta(s)/Zeta(4s). - Enrique Pérez Herrero, Jan 21 2013
A051903(a(n)) > 3. - Reinhard Zumkeller, Sep 03 2015
The asymptotic density of this sequence is 1 - 1/zeta(4) = 1 - 90/Pi^4 = 0.076061... - Amiram Eldar, Jul 09 2020
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Biquadratefree.
MAPLE
with(NumberTheory):
isBiquadrateful := n -> is(denom(Radical(n) / LargestNthPower(n, 2)) <> 1):
select(isBiquadrateful, [`$`(1..704)]); # Peter Luschny, Jul 12 2022
MATHEMATICA
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 *)
PROG
(Haskell)
a046101 n = a046101_list !! (n-1)
a046101_list = filter ((> 3) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015
(PARI) is(n)=n>9 && vecmax(factor(n)[, 2])>3 \\ Charles R Greathouse IV, Sep 03 2015
(Python)
from sympy import mobius, integer_nthroot
def A046101(n):
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)
return m # Chai Wah Wu, Aug 05 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved