A062320 Nonsquarefree numbers squared. A013929(n)^2.

16, 64, 81, 144, 256, 324, 400, 576, 625, 729, 784, 1024, 1296, 1600, 1936, 2025, 2304, 2401, 2500, 2704, 2916, 3136, 3600, 3969, 4096, 4624, 5184, 5625, 5776, 6400, 6561, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456

Offset: 1

Author

Jason Earls, Jul 05 2001

Keywords

• easy
• nonn

Comments

A008966(A037213(a(n))) = 0. - Reinhard Zumkeller, Sep 03 2015

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A013929, A008966, A037213, A320965.

Squares in A046099.

Programs

Haskell

a062320 = (^ 2) . a013929 -- Reinhard Zumkeller, Sep 03 2015

PARI

for(n=1,55, if(issquarefree(n), n+1,print(n^2)))

PARI

n=-1; for (m=1, 10^9, if (!issquarefree(m), write("b062320.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 04 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 A062320(n):
    def f(x): return n+1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 31 2024

Formula

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2. - Amiram Eldar, Jul 16 2020

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 11 2001

Offset corrected by Andrew Howroyd, Sep 18 2024