A132295 Sum of the nonsquare numbers not larger than n.

0, 2, 5, 5, 10, 16, 23, 31, 31, 41, 52, 64, 77, 91, 106, 106, 123, 141, 160, 180, 201, 223, 246, 270, 270, 296, 323, 351, 380, 410, 441, 473, 506, 540, 575, 575, 612, 650, 689, 729, 770, 812, 855, 899, 944, 990, 1037, 1085, 1085, 1135, 1186, 1238, 1291, 1345

Offset

1,2

Links

Jason Yuen, Table of n, a(n) for n = 1..10000

Formula

Let r = floor(sqrt(n)). Then a(n) = n*(n+1)/2 - r*(r+1)*(2*r+1)/6.

Example

Let n=5. The sum of the nonsquare numbers <= 5 is 2+3+5 = 10, the 5th entry in the sequence.

Mathematica

Table[Total[Select[Range[n], !IntegerQ[Sqrt[#]]&]], {n, 54}] (* James C. McMahon, Mar 04 2025 *)

Prog

(PARI) a(n)= { my(r=sqrtint(n), sq=r*(r+1)*(2*r+1)/6, sn=n*(n+1)/2); sn-sq }
(Python)
from math import isqrt
def A000330(n): return n*(n+1)*(2*n+1)//6
def A132295(n): return n*(n+1)//2-A000330(isqrt(n)) # Jason Yuen, Jan 30 2024

Crossrefs

Cf. A000037, A000330, A086849.

Sequence in context: A081240 A184443 A238655 * A086651 A074495 A081467

Adjacent sequences: A132292 A132293 A132294 * A132296 A132297 A132298

Keyword

nonn,easy

Author

Cino Hilliard, Nov 07 2007

Extensions

Definition corrected by R. J. Mathar, Sep 10 2016

Status

approved