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A066779 Sum of squarefree numbers <= n.

%I A066779 #37 Mar 07 2021 12:17:43
%S A066779 1,3,6,6,11,17,24,24,24,34,45,45,58,72,87,87,104,104,123,123,144,166,
%T A066779 189,189,189,215,215,215,244,274,305,305,338,372,407,407,444,482,521,
%U A066779 521,562,604,647,647,647,693,740,740,740,740,791,791,844,844,899,899
%N A066779 Sum of squarefree numbers <= n.
%D A066779 D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.
%H A066779 Harry J. Smith, <a href="/A066779/b066779.txt">Table of n, a(n) for n = 1..1000</a>
%F A066779 a(n) = Sum_{i=1..n} mu(i)^2*i.
%F A066779 a(n) = Sum_{k=1..n} k*A008966(k). - _Reinhard Zumkeller_, Jul 05 2010
%F A066779 a(n) = Sum_{d=1..sqrt(n)} mu(d)*d^2*floor(n/d^2)*floor(n/d^2+1)/2. - _Charles R Greathouse IV_, Apr 26 2012
%F A066779 G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x). - _Ilya Gutkovskiy_, Apr 16 2017
%F A066779 a(n) ~ (3/Pi^2) * n^2 + O(n^(3/2)) (Suryanarayana, 1969). - _Amiram Eldar_, Mar 07 2021
%t A066779 Table[ n*Boole[ SquareFreeQ[n] ], {n, 1, 56}] // Accumulate (* _Jean-François Alcover_, Jun 18 2013 *)
%o A066779 (PARI) s=0; for (n=1, 1000, write("b066779.txt", n, " ", s+=moebius(n)^2*n) ) \\ _Harry J. Smith_, Mar 24 2010
%o A066779 (PARI) a(n)=sum(d=1,sqrtint(n),moebius(d)*d^2*binomial(n\d^2+1,2)) \\ _Charles R Greathouse IV_, Apr 26 2012
%o A066779 (PARI) a(n)=my(s,k2); forsquarefree(k=1,sqrtint(n), k2=k[1]^2; s+= k2*binomial(n\k2+1,2)*moebius(k)); s \\ _Charles R Greathouse IV_, Jan 08 2018
%o A066779 (Python)
%o A066779 from sympy.ntheory.factor_  import core
%o A066779 def a(n): return sum ([i for i in range(1, n + 1) if core(i) == i]) # _Indranil Ghosh_, Apr 16 2017
%Y A066779 Cf. A008966, A013928, A034387, A179213, A179215.
%K A066779 nonn,easy
%O A066779 1,2
%A A066779 _Benoit Cloitre_, Jan 18 2002