A086849 Sum of first n nonsquares.

2, 5, 10, 16, 23, 31, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 223, 246, 270, 296, 323, 351, 380, 410, 441, 473, 506, 540, 575, 612, 650, 689, 729, 770, 812, 855, 899, 944, 990, 1037, 1085, 1135, 1186, 1238, 1291, 1345, 1400, 1456, 1513, 1571, 1630

Offset: 1

Author

Reinhard Zumkeller, Aug 18 2003

Keywords

• nonn
• easy

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000037, A000196, A000217, A000330, A048395.

Programs

Haskell

a086849 n = a086849_list !! (n-1)
a086849_list = scanl1 (+) a000037_list
-- Reinhard Zumkeller, Oct 26 2015

Mathematica

Accumulate[Table[n + Round[Sqrt[n]], {n, 120}]] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
Accumulate[DeleteCases[Range[80],?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale, Jun 11 2024 *)

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

Python

from math import isqrt
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

Formula

a(n) = A000217(A000037(n)) - A000330(A000196(A000037(n))).

From Jonathan Vos Post, Aug 28 2005: (Start)

a(n) = Sum_{i=1..n} A000037(i).

a(n) = Sum_{i=1..n} (i + floor(1/2 + sqrt(i))). (End)

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

a(n) = n^2/2 + 2n*sqrt(n)/3 + O(n). - Charles R Greathouse IV, Aug 28 2016