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A046100 Biquadratefree numbers. 36
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76 (list; graph; refs; listen; history; text; internal format)
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
CROSSREFS
Cf. A046101, A005117 (2-free), A004709 (3-free).
Cf. A051903, subsequence of A209061.
Sequence in context: A013938 A339889 A023809 * A337138 A023757 A115105
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified June 17 15:57 EDT 2024. Contains 373463 sequences. (Running on oeis4.)