A340588 Squares of perfect powers.

1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 6561, 10000, 14641, 15625, 16384, 20736, 28561, 38416, 46656, 50625, 59049, 65536, 83521, 104976, 117649, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1000000

Offset: 1

Author

Terry D. Grant, Sep 21 2020

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A001597, A062965, A072102, A117453, A131605.

Cf. A153158 (complement within positive squares).

Programs

Maple

q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))<>2):
select(q, [i^2$i=1..1000])[];  # Alois P. Heinz, Nov 26 2024

Mathematica

Join[{1}, (Select[Range[2000], GCD @@ FactorInteger[#][[All, 2]] > 1 &])^2]

Python

from sympy import mobius, integer_nthroot
def A340588(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    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 14 2024

Formula

a(n) = A001597(n)^2.

a(n+1) = A062965(n) + 1. - Hugo Pfoertner, Sep 29 2020

Sum_{k>1} 1/(a(k) - 1) = 7/4 - Pi^2/6 = 7/4 - zeta(2).

Sum_{k>1} 1/a(k) = Sum_{k>=2} mu(k)*(1-zeta(2*k)).