OFFSET
1,1
COMMENTS
It appears that all terms of this sequence are exactly four times those of the squarefree integers (A005117).
The observed behavior is true for all n. All positive integers n are written uniquely as k*m^2 where k is squarefree, k >=1, m >= 1. The square multiples of n are j^2*k*n, j >= 1. We seek n with exactly 1 multiple that is square and less than n^2. If m = 1, there are no such multiples as we have k = n, so the least square multiple is n^2. If m >= 2, k*n is square and less than n^2. However, 4*k*n also qualifies as square and less than n^2 if m > 2. So the qualifying values of n are those with m=2. - Peter Munn, Nov 28 2019
The asymptotic density of this sequence is 3/(2*Pi^2). - Amiram Eldar, Mar 08 2021
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
FORMULA
A057918(a(n)) = 1. - Reinhard Zumkeller, Mar 27 2012
From Peter Munn, Nov 28 2019: (Start)
a(n) = 4 * A005117(n).
(End)
Sum_{n>=1} 1/a(n)^s = zeta(s)/(4^s*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023
EXAMPLE
4 is in the sequence because among the products 1*4,2*4,3*4 = 4,8,12 there is exactly one square.
MATHEMATICA
eoiQ[n_]:=Count[n*Range[n-1], _?(IntegerQ[Sqrt[#]]&)]==1; Select[Range[ 400], eoiQ] (* Harvey P. Dale, Mar 14 2015 *)
PROG
(Haskell)
a133466 n = a133466_list !! (n-1)
a133466_list = map (+ 1) $ elemIndices 1 a057918_list
-- Reinhard Zumkeller, Mar 27 2012
(PARI) isok(n) = sum(k=1, n-1, issquare(k*n)) == 1; \\ Michel Marcus, Nov 29 2019
(Magma) [k:k in [1..350]|#[m:m in [1..k-1]| IsSquare(m*k)] eq 1]; // Marius A. Burtea, Dec 03 2019
(Python)
from math import isqrt
from sympy import mobius
def A133466(n):
def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
return int(m)<<2 # Chai Wah Wu, Aug 15 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
John W. Layman, Nov 28 2007
STATUS
approved