A046100 Biquadratefree numbers.

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

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

Adjacent sequences: A046097 A046098 A046099 * A046101 A046102 A046103

Keyword

nonn

Author

Eric W. Weisstein

Status

approved