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A380857 Squares of numbers that are neither squarefree nor prime powers.

%I A380857 #16 Feb 08 2025 23:34:27
%S A380857 144,324,400,576,784,1296,1600,1936,2025,2304,2500,2704,2916,3136,
%T A380857 3600,3969,4624,5184,5625,5776,6400,7056,7744,8100,8464,9216,9604,
%U A380857 9801,10000,10816,11664,12544,13456,13689,14400,15376,15876,17424,18225,18496,19600,20736
%N A380857 Squares of numbers that are neither squarefree nor prime powers.
%C A380857 Proper subset of A359280 which is a proper subset of A286708 (powerful numbers that are not prime powers, a proper subset of A126706).
%C A380857 Does not intersect A362605.
%H A380857 Michael De Vlieger, <a href="/A380857/b380857.txt">Table of n, a(n) for n = 1..10000</a>
%F A380857 a(n) = A126706(n)^2.
%F A380857 Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2 - Sum_{p prime} 1/(p^2*(p^2-1)) = A013661 - A082020 + A085548 - A154945 = 0.025670434597226178881... . - _Amiram Eldar_, Feb 08 2025
%t A380857 Select[Range[150], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]^2
%o A380857 (PARI) isok(k) = !issquarefree(k) && !isprimepower(k); \\ A126706
%o A380857 apply(sqr, select(isok, [1..200])) \\ _Michel Marcus_, Feb 07 2025
%o A380857 (Python)
%o A380857 from math import isqrt
%o A380857 from sympy import primepi, integer_nthroot, mobius
%o A380857 def A380857(n):
%o A380857     def bisection(f,kmin=0,kmax=1):
%o A380857         while f(kmax) > kmax: kmax <<= 1
%o A380857         kmin = kmax >> 1
%o A380857         while kmax-kmin > 1:
%o A380857             kmid = kmax+kmin>>1
%o A380857             if f(kmid) <= kmid:
%o A380857                 kmax = kmid
%o A380857             else:
%o A380857                 kmin = kmid
%o A380857         return kmax
%o A380857     def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A380857     return bisection(f,n,n)**2 # _Chai Wah Wu_, Feb 08 2025
%Y A380857 Cf. A059404, A126706, A177492 (k^2 for k in A120944), A286708, A359280, A362605, A378768 (k^2 for k in A286708).
%Y A380857 Cf. A013661, A082020, A085548, A154945.
%K A380857 nonn,easy
%O A380857 1,1
%A A380857 _Michael De Vlieger_, Feb 06 2025