A062965 -id:A062965 - cp's OEIS frontend

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

Terry D. Grant, Sep 21 2020

nonn

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

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

Cf. A153158 (complement within positive squares).

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

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

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

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)).

A062757 Denominator of sum of first n terms of the series 1/15 + 1/63 + 1/80 ... in which the denominators are perfect squares - 1 which are simultaneously other powers, e.g. a(1) = 15 because 16 = 4^2 = 2^4, a perfect square that is also a fourth power; hence 16-1 = 15 qualifies as a term.

Original entry on oeis.org

15, 315, 5040, 85680, 278460, 42840, 14608440, 540512280, 10810245600, 46844397600, 480155075400, 145486987846200, 17749412517236400, 5916470839078800, 10769949084069775600, 312328523438023492400

Offset: 1

Jason Earls, Jul 16 2001

nonn

			a(2)=63 because the perfect square 64= 8^2 = 4^3.

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington D.C., 1999, p. 65.
L. Euler, "Variae observationes circa series infinitas," Opera Omnia, Ser. 1, Vol. 14, pp. 216-244.

L. Euler, Variae observationes circa series infinitas

Cf. A037450, A062834, A062965, A001597.

Table[ Denominator[ Plus@@(Take[ Select[ Range[ 2, 150 ], GCD@@(Last/@FactorInteger[ # ])>1& ]^2-1, k ]^-1) ], {k, 1, 16} ]

More terms from Dean Hickerson, Jul 24 2001

cp's OEIS Frontend

A340588 Squares of perfect powers.

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