A046100 Biquadratefree numbers: numbers that are not divisible by any 4th power greater than 1.

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

Author

Eric W. Weisstein

Keywords

• nonn

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.

Crossrefs

Cf. A046101, A005117 (2-free), A004709 (3-free).

Cf. A023809, A051903, A215267.

Subsequence of A209061.

Programs

Haskell

a046100 n = a046100_list !! (n-1)
a046100_list = filter ((< 4) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015

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 *)

PARI

is(n)=n==1 || vecmax(factor(n)[,2])<4 \\ Charles R Greathouse IV, Jun 16 2012

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

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

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

Extensions

Name edited by Amiram Eldar, Jul 29 2024