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

1,1

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

Crossrefs

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

Sequence in context: A359831 A048111 A122614 * A360723 A044856 A217558

Adjacent sequences: A046098 A046099 A046100 * A046102 A046103 A046104

Keyword

nonn

Author

Eric W. Weisstein

Status

approved