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A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

%I A384517 #15 Jun 01 2025 16:38:06
%S A384517 4,9,16,25,36,49,64,81,100,121,169,196,225,256,289,361,441,484,529,
%T A384517 625,676,729,841,900,961,1024,1089,1156,1225,1296,1369,1444,1521,1681,
%U A384517 1764,1849,2116,2209,2401,2601,2809,3025,3249,3364,3481,3721,3844,4096,4225,4356
%N A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.
%C A384517 Differs from its subsequence A340674 by having the terms 64, 729, 1024, 4096, .... .
%C A384517 Numbers whose prime factorization exponents are equal and even.
%H A384517 Amiram Eldar, <a href="/A384517/b384517.txt">Table of n, a(n) for n = 1..10000</a>
%H A384517 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A384517 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A384517 a(n) = A062770(n)^2 = A072774(n+1)^2.
%F A384517 Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k)/zeta(4*k)-1) = Sum{k>=1} (A231327(k)/(A231273(k)*Pi^(2*k)) - 1) = 0.62022193512079649421... .
%t A384517 Select[Range[2, 100], SameQ @@ FactorInteger[#][[;;, 2]] &]^2
%o A384517 (PARI) isok(k) = {my(s, e = ispower(k, , &s)); !(e % 2) && issquarefree(s);}
%o A384517 (Python)
%o A384517 from math import isqrt
%o A384517 from sympy import mobius, integer_nthroot
%o A384517 def A384517(n):
%o A384517     def bisection(f,kmin=0,kmax=1):
%o A384517         while f(kmax) > kmax: kmax <<= 1
%o A384517         kmin = kmax >> 1		
%o A384517         while kmax-kmin > 1:
%o A384517             kmid = kmax+kmin>>1
%o A384517             if f(kmid) <= kmid:
%o A384517                 kmax = kmid
%o A384517             else:
%o A384517                 kmin = kmid
%o A384517         return kmax
%o A384517     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A384517     def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(2,x.bit_length(),2))
%o A384517     return bisection(f,n,n) # _Chai Wah Wu_, Jun 01 2025
%Y A384517 Intersection of A000290 and A072777.
%Y A384517 Equals A072777 \ A384518.
%Y A384517 A340674 is a subsequence.
%Y A384517 Cf. A005117, A062770, A072774, A231273, A231327.
%K A384517 nonn
%O A384517 1,1
%A A384517 _Amiram Eldar_, Jun 01 2025