A382967 Biquadratefree numbers (A046100) that are not squarefree (A005117).

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 84, 88, 90, 92, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 180, 184

Offset

1,1

Comments

Subsequence of A252849 and first differs from it at n = 22: A252849(22) = 64 = 2^6 is not a term of this sequence.

Subsequence of A375229 and differs from it by not having the terms 1, 256, 512, 768, 1024, ... .

Numbers whose prime factorization has least one exponent that equals 2 or 3 and no higher exponent.

Numbers k such that 2 <= A051903(k) <= 3.

The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(2) = A215267 - A059956 = 0.3160113... .

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (12*(15 - Pi^2)). - Vaclav Kotesovec, Apr 11 2025

Mathematica

Select[Range[200], 2 <= Max[FactorInteger[#][[;; , 2]]] <= 3 &]

Prog

(PARI) isok(k) = if(k == 1, 0, my(emax = vecmax(factor(k)[, 2])); emax > 1 && emax < 4);
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A382967(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+sum(mobius(k)*(x//k**2-x//k**4) for k in range(1, integer_nthroot(x, 4)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x, 4)[0]+1, isqrt(x)+1)))
    return bisection(f, n, n) # Chai Wah Wu, Apr 11 2025

Crossrefs

Disjoint union of A067259 and A375072.

Intersection of A046100 and A013929.

Subsequence of A252849 and A375229.

Cf. A004709, A005117, A051903, A059956, A215267.

Sequence in context: A370076 A252849 A375229 * A082293 A374593 A048944

Adjacent sequences: A382964 A382965 A382966 * A382968 A382969 A382970

Keyword

nonn,easy

Author

Amiram Eldar, Apr 10 2025

Status

approved