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A071404 Which nonsquarefree number is a square number? a(n)-th nonsquarefree number equals n^2, the n-th square.

%I A071404 #35 Jul 21 2024 03:34:30
%S A071404 1,3,5,9,13,18,25,31,39,46,55,66,76,86,99,112,125,142,157,172,187,206,
%T A071404 225,244,264,285,307,328,353,377,400,429,453,480,507,534,562,593,623,
%U A071404 656,691,725,762,795,831,867,904,941,977,1019,1059,1101,1145,1185,1226
%N A071404 Which nonsquarefree number is a square number? a(n)-th nonsquarefree number equals n^2, the n-th square.
%H A071404 Amiram Eldar, <a href="/A071404/b071404.txt">Table of n, a(n) for n = 2..10001</a>
%F A071404 A013929(a(n)) = A000290(n) = n^2.
%F A071404 a(n) = kn^2 + O(n), where k = 1 - 6/Pi^2. - _Charles R Greathouse IV_, Sep 13 2013
%F A071404 a(n) = -Sum_{k=2..n} mu(k)*floor((n/k)^2). - _Chai Wah Wu_, Jul 20 2024
%e A071404 The first, 3rd, 5th, 9th, and 13th nonsquarefree numbers are 4, 9, 16, 25, and 36, respectively.
%t A071404 Position[Select[Range[3200], !SquareFreeQ[#] &], _?(IntegerQ[Sqrt[#]] &)][[;; , 1]] (* _Amiram Eldar_, Mar 04 2024 *)
%o A071404 (PARI) lista(nn) = {sqfs = select(n->(1-issquarefree(n)), vector(nn, i, i)); for (i = 1, #sqfs, if (issquare(sqfs[i]), print1(i, ", ")););} \\ _Michel Marcus_, Sep 12 2013
%o A071404 (PARI) a(n)=n^2-sum(k=1, n, n^2\k^2*moebius(k)) \\ _Charles R Greathouse IV_, Sep 13 2013
%o A071404 (Python)
%o A071404 from sympy import mobius
%o A071404 def A071404(n): return -sum(mobius(k)*(n**2//k**2) for k in range(2,n+1)) # _Chai Wah Wu_, Jul 20 2024
%Y A071404 Cf. A005117, A013929, A000040, A000290, A071403.
%K A071404 nonn
%O A071404 2,2
%A A071404 _Labos Elemer_, May 24 2002
%E A071404 Offset changed by _Charles R Greathouse IV_, Sep 13 2013