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A087050 Square root of the largest square >1 dividing the n-th nonsquarefree number.

%I A087050 #21 Jul 23 2024 10:51:23
%S A087050 2,2,3,2,4,3,2,2,5,3,2,4,6,2,2,3,4,7,5,2,3,2,2,3,8,2,6,5,2,4,9,2,2,3,
%T A087050 2,4,7,3,10,2,6,4,2,3,2,11,2,5,3,8,2,3,2,2,12,7,2,5,2,3,2,4,9,2,2,13,
%U A087050 3,2,5,4,6,2,2,3,8,14,3,10,2,3,4,2,6,2,4,15,2,2,3,2,4,11,9,2,7,2,5,6,16,2,3
%N A087050 Square root of the largest square >1 dividing the n-th nonsquarefree number.
%H A087050 Amiram Eldar, <a href="/A087050/b087050.txt">Table of n, a(n) for n = 1..10000</a>
%F A087050 a(n)^2 is the largest square factor (from A000290) of the nonsquarefree number A013929(n), n>=1.
%F A087050 a(n) = A000188(A013929(n)). - _Amiram Eldar_, Feb 11 2021
%F A087050 Sum_{k=1..n} a(k) ~ (n/(2*(zeta(2)-1))) * (log(n) + 3*gamma - 3 - 2*zeta'(2)/zeta(2) - log(1-1/zeta(2))), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Jan 14 2024
%e A087050 n=10, A013929(10) = 27, a(10)^2 = 3^2 = 9. 27 = 9*3.
%e A087050 n=39, A013929(39) = 100, a(39)^2 = 10^2 = 100.
%t A087050 f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[300], !SquareFreeQ[#] &] (* _Amiram Eldar_, Feb 11 2021 *)
%o A087050 (Python)
%o A087050 from math import isqrt, prod
%o A087050 from sympy import mobius, factorint
%o A087050 def A087050(n):
%o A087050     def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A087050     m, k = n, f(n)
%o A087050     while m != k:
%o A087050         m, k = k, f(k)
%o A087050     return prod(p**(e>>1) for p, e in factorint(m).items() if e>1) # _Chai Wah Wu_, Jul 22 2024
%Y A087050 Cf. A087049, A013929, A000188, A000290.
%Y A087050 Cf. A001610, A013661, A306016.
%K A087050 nonn,easy
%O A087050 1,1
%A A087050 _Wolfdieter Lang_, Sep 08 2003