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Squares of perfect powers.
3

%I #26 Nov 26 2024 15:59:22

%S 1,16,64,81,256,625,729,1024,1296,2401,4096,6561,10000,14641,15625,

%T 16384,20736,28561,38416,46656,50625,59049,65536,83521,104976,117649,

%U 130321,160000,194481,234256,262144,279841,331776,390625,456976,531441,614656,707281,810000,923521,1000000

%N Squares of perfect powers.

%H Amiram Eldar, <a href="/A340588/b340588.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n+1) = A062965(n) + 1. - _Hugo Pfoertner_, Sep 29 2020

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

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

%p q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))<>2):

%p select(q, [i^2$i=1..1000])[]; # _Alois P. Heinz_, Nov 26 2024

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

%o (Python)

%o from sympy import mobius, integer_nthroot

%o def A340588(n):

%o 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())))

%o kmin, kmax = 1,2

%o while f(kmax) >= kmax:

%o kmax <<= 1

%o while True:

%o kmid = kmax+kmin>>1

%o if f(kmid) < kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o if kmax-kmin <= 1:

%o break

%o return kmax**2 # _Chai Wah Wu_, Aug 14 2024

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

%Y Cf. A153158 (complement within positive squares).

%K nonn

%O 1,2

%A _Terry D. Grant_, Sep 21 2020