A014198 Number of integer solutions to x^2 + y^2 <= n excluding (0,0).

0, 4, 8, 8, 12, 20, 20, 20, 24, 28, 36, 36, 36, 44, 44, 44, 48, 56, 60, 60, 68, 68, 68, 68, 68, 80, 88, 88, 88, 96, 96, 96, 100, 100, 108, 108, 112, 120, 120, 120, 128, 136, 136, 136, 136, 144, 144, 144, 144, 148, 160, 160, 168, 176, 176, 176, 176, 176, 184, 184

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(32)/32 = 100/32 = 3.125; lim_{n->infinity} a(n)/n = Pi.

The terms of this sequence are four times the running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors. - Ant King, Mar 12 2013

Examples

			For n=2 the 8 solutions are (x,y) = (+-1,0), (0,+-1), (+-1,+-1).
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 339

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Sum of Squares Function

Crossrefs

Cf. A014200, A057655.

Programs

Maple

A014198 := proc(n)
    nops([ numtheory[thue]( abs( x^2+y^2) <= n, [ x, y ] ) ]);
end proc:
seq(A014198(n),n=0..60) ;

Mathematica

Prepend[SquaresR[2,#] &/@Range[59],0]//Accumulate (* Ant King, Mar 12 2013 *)

PARI

a(n)=local(j); j=sqrtint(n); sum(x=-j,j,sum(y=-j,j,x^2+y^2<=n))-1

Python

from math import prod
from itertools import count, accumulate, islice
from sympy import factorint
def A014198_gen(): # generator of terms
    return accumulate(map(lambda n:prod(e+1 if p & 3 == 1 else (e+1) & 1 for p, e in factorint(n).items() if p > 2) << 2, count(1)),initial=0)
A014198_list = list(islice(A014198_gen(),30)) # Chai Wah Wu, Jun 28 2022

Formula

a(n) = 4*A014200(n).

a(n) = A057655(n)-1.