A046101 Biquadrateful numbers.

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

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

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.

Crossrefs

Cf. A046100, A046099, A036967, A001694, A051903, A215267.

Programs

Haskell

a046101 n = a046101_list !! (n-1)
a046101_list = filter ((> 3) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015

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 *)

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