OFFSET
1,2
COMMENTS
Differs from A023809 at entries 0, 81, 162, 225, 226, etc. - R. J. Mathar, Oct 18 2008
Density is 1/zeta(4) = A215267 = 0.923938.... - Charles R Greathouse IV, Sep 02 2015
The Schnirelmann density of the biquadratefree numbers is 145/157 (Orr, 1969). - Amiram Eldar, Mar 12 2021
This sequence has arbitrarily large gaps and hence is not a Beatty sequence. - Charles R Greathouse IV, Jan 27 2022
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
Eric Weisstein's World of Mathematics, Biquadratefree.
FORMULA
A051903(a(n)) < 4. - Reinhard Zumkeller, Sep 03 2015
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(4*s), for s > 1. - Amiram Eldar, Dec 27 2022
MAPLE
A046100 := proc(n)
option remember;
local a, p, is4free;
if n = 1 then
return 1;
else
for a from procname(n-1)+1 do
is4free := true ;
for p in ifactors(a)[2] do
if op(2, p) >= 4 then
is4free := false;
break;
end if;
end do:
if is4free then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, Aug 08 2012
MATHEMATICA
lst={}; Do[a=0; Do[If[FactorInteger[m][[n, 2]]>4, a=1], {n, Length[FactorInteger[m]]}]; If[a!=1, AppendTo[lst, m]], {m, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
Select[Range[100], Max[FactorInteger[#][[;; , 2]]]<4&] (* Harvey P. Dale, Jul 13 2023 *)
PROG
(PARI) is(n)=n==1 || vecmax(factor(n)[, 2])<4 \\ Charles R Greathouse IV, Jun 16 2012
(Sage)
def is_biquadratefree(n):
return all(c[1] < 4 for c in n.factor())
def A046100_list(n): return [i for i in (1..n) if is_biquadratefree(i)]
A046100_list(76) # Peter Luschny, Aug 08 2012
(Haskell)
a046100 n = a046100_list !! (n-1)
a046100_list = filter ((< 4) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015
(Python)
from sympy import mobius, integer_nthroot
def A046100(n):
def f(x): return n+x-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
EXTENSIONS
Name edited by Amiram Eldar, Jul 29 2024
STATUS
approved