A062503 Squarefree numbers squared.

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476

Offset: 1

Author

Jason Earls, Jul 09 2001

Keywords

• nonn

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

Crossrefs

Characteristic function is A227291.

Other powers of squarefree numbers: A005117(1), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7), A072774(all).

Cf. A000290, A001694, A052485, A062320.

Cf. A001248 (a subsequence).

A329332 column 2 in ascending order.

Programs

Haskell

a062503 = a000290 . a005117  -- Reinhard Zumkeller, Jul 07 2013

Mathematica

Select[Range[100], SquareFreeQ]^2

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

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

Formula

Numbers k such that Sum_{d|k} mu(d)*mu(k/d) = 1. - Benoit Cloitre, Mar 03 2004

a(n) = A000290(A005117(n)); A227291(a(n)) = 1. - Reinhard Zumkeller, Jul 07 2013

A000290 \ A062320. - R. J. Mathar, Jul 27 2013

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