A082293 Numbers having exactly one square divisor > 1.

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 75, 76, 84, 88, 90, 92, 98, 99, 104, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 184, 188, 189, 198, 204, 207, 212

Offset: 1

Author

Reinhard Zumkeller, Apr 08 2003

Keywords

• nonn

Comments

Numbers of the form m*p^2, p prime and m squarefree (A005117). [Corrected by Peter Munn, Nov 17 2020]

The asymptotic density of this sequence is (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2 = 0.274933... (A222056). - Amiram Eldar, Jul 07 2020

Links

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

Crossrefs

Cf. A005117, A046951, A222056.

Complement of A048111 within A013929.

Subsequence of A252849.

Disjoint union of A048109 and A060687.

A285508 is a subsequence.

Programs

Mathematica

Select[Range[2, 200], MemberQ[{2, 3}, (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]]] && (Length[e] == 1 || e[[-2]] == 1) &] (* Amiram Eldar, Jul 07 2020 *)

PARI

is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && f[1]>1 && f[1]<4 && (#f==1 || f[2]==1) \\ Charles R Greathouse IV, Oct 16 2015

Python

from math import isqrt
from sympy import mobius, primerange
def A082293(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 g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x-sum(g(x//p**2) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

Formula

A046951(a(n)) = 2.