OFFSET
1,2
COMMENTS
Also, except for the initial term, numbers whose prime factors are squared. - Cino Hilliard, Jan 25 2006
Also cubefree numbers that are squares. - Gionata Neri, May 08 2016
All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (term of this sequence), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on. - Peter Munn, Mar 12 2020
Powerful numbers (A001694) all of whose nonunitary divisors are not powerful (A052485). - Amiram Eldar, May 13 2023
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
FORMULA
Numbers k such that Sum_{d|k} mu(d)*mu(k/d) = 1. - Benoit Cloitre, Mar 03 2004
a(n) ~ (Pi^4/36) * n^2. - Charles R Greathouse IV, Nov 24 2015
a(n) = A046692(a(n))^2. - Torlach Rush, Jan 05 2019
For all k in the sequence, Omega(k) = 2*omega(k). - Wesley Ivan Hurt, Apr 30 2020
Sum_{n>=1} 1/a(n) = zeta(2)/zeta(4) = 15/Pi^2 (A082020). - Amiram Eldar, May 22 2020
MATHEMATICA
Select[Range[100], SquareFreeQ]^2
PROG
(PARI) je=[]; for(n=1, 200, if(issquarefree(n), je=concat(je, n^2), )); je
(PARI) n=0; for (m=1, 10^5, if(issquarefree(m), write("b062503.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 08 2009
(PARI) is(n)=issquare(n, &n) && issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015
(Haskell)
a062503 = a000290 . a005117 -- Reinhard Zumkeller, Jul 07 2013
(Python)
from math import isqrt
from sympy import mobius
def A062503(n):
def f(x): return n-1+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
kmin, kmax = 1, 2
while f(kmax) >= kmax:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if f(kmid) < kmid:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
return kmax**2 # Chai Wah Wu, Aug 19 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Jason Earls, Jul 09 2001
STATUS
approved