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A340588 Squares of perfect powers.

%I A340588 #26 Nov 26 2024 15:59:22
%S A340588 1,16,64,81,256,625,729,1024,1296,2401,4096,6561,10000,14641,15625,
%T A340588 16384,20736,28561,38416,46656,50625,59049,65536,83521,104976,117649,
%U A340588 130321,160000,194481,234256,262144,279841,331776,390625,456976,531441,614656,707281,810000,923521,1000000
%N A340588 Squares of perfect powers.
%H A340588 Amiram Eldar, <a href="/A340588/b340588.txt">Table of n, a(n) for n = 1..10000</a>
%F A340588 a(n) = A001597(n)^2.
%F A340588 a(n+1) = A062965(n) + 1. - _Hugo Pfoertner_, Sep 29 2020
%F A340588 Sum_{k>1} 1/(a(k) - 1) = 7/4 - Pi^2/6 = 7/4 - zeta(2).
%F A340588 Sum_{k>1} 1/a(k) = Sum_{k>=2} mu(k)*(1-zeta(2*k)).
%p A340588 q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))<>2):
%p A340588 select(q, [i^2$i=1..1000])[];  # _Alois P. Heinz_, Nov 26 2024
%t A340588 Join[{1}, (Select[Range[2000], GCD @@ FactorInteger[#][[All, 2]] > 1 &])^2]
%o A340588 (Python)
%o A340588 from sympy import mobius, integer_nthroot
%o A340588 def A340588(n):
%o A340588     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 A340588     kmin, kmax = 1,2
%o A340588     while f(kmax) >= kmax:
%o A340588         kmax <<= 1
%o A340588     while True:
%o A340588         kmid = kmax+kmin>>1
%o A340588         if f(kmid) < kmid:
%o A340588             kmax = kmid
%o A340588         else:
%o A340588             kmin = kmid
%o A340588         if kmax-kmin <= 1:
%o A340588             break
%o A340588     return kmax**2 # _Chai Wah Wu_, Aug 14 2024
%Y A340588 Cf. A000290, A001597, A062965, A072102, A117453, A131605.
%Y A340588 Cf. A153158 (complement within positive squares).
%K A340588 nonn
%O A340588 1,2
%A A340588 _Terry D. Grant_, Sep 21 2020