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

1,1

Comments

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

Links

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

Formula

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

Prog

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

Crossrefs

Cf. A013929, A008966, A037213, A320965.

Squares in A046099.

Sequence in context: A092210 A320892 A365263 * A233330 A370787 A322449

Adjacent sequences: A062317 A062318 A062319 * A062321 A062322 A062323

Keyword

easy,nonn

Author

Jason Earls, Jul 05 2001

Extensions

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

Offset corrected by Andrew Howroyd, Sep 18 2024

Status

approved