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A086849 Sum of first n nonsquares.

%I A086849 #41 Mar 08 2025 14:57:52
%S A086849 2,5,10,16,23,31,41,52,64,77,91,106,123,141,160,180,201,223,246,270,
%T A086849 296,323,351,380,410,441,473,506,540,575,612,650,689,729,770,812,855,
%U A086849 899,944,990,1037,1085,1135,1186,1238,1291,1345,1400,1456,1513,1571,1630
%N A086849 Sum of first n nonsquares.
%D A086849 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.
%H A086849 Reinhard Zumkeller, <a href="/A086849/b086849.txt">Table of n, a(n) for n = 1..10000</a>
%F A086849 a(n) = A000217(A000037(n)) - A000330(A000196(A000037(n))).
%F A086849 From _Jonathan Vos Post_, Aug 28 2005: (Start)
%F A086849 a(n) = Sum_{i=1..n} A000037(i).
%F A086849 a(n) = Sum_{i=1..n} (i + floor(1/2 + sqrt(i))). (End)
%F A086849 a(n) = floor(1/2 + (n + sqrt(n))*(n/2 + sqrt(n)/6 + 1/3) - (floor(1/2 + sqrt(n)) - sqrt(n))^2*sqrt(n)). - _Graeme McRae_, Aug 28 2007
%F A086849 a(n) = n^2/2 + 2n*sqrt(n)/3 + O(n). - _Charles R Greathouse IV_, Aug 28 2016
%t A086849 Accumulate[Table[n + Round[Sqrt[n]], {n, 120}]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011 *)
%t A086849 Accumulate[DeleteCases[Range[80],_?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale_, Jun 11 2024 *)
%o A086849 (Haskell)
%o A086849 a086849 n = a086849_list !! (n-1)
%o A086849 a086849_list = scanl1 (+) a000037_list
%o A086849 -- _Reinhard Zumkeller_, Oct 26 2015
%o A086849 (PARI) a(n)=my(k=n+(sqrtint(4*n)+1)\2,s=sqrtint(k)); k*(k+1)/2 - s*(s+1)*(2*s+1)/6 \\ _Charles R Greathouse IV_, Aug 28 2016
%o A086849 (Python)
%o A086849 from math import isqrt
%o A086849 def A086849(n): return (m:= n + isqrt(n + isqrt(n)))*(m + 1)//2 - (k:=isqrt(m))*(k + 1)*(2*k + 1)//6 # _Chai Wah Wu_, Mar 31 2022
%Y A086849 Cf. A000037, A000196, A000217, A000330, A048395.
%K A086849 nonn,easy
%O A086849 1,1
%A A086849 _Reinhard Zumkeller_, Aug 18 2003