A014198 - OEIS

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

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OFFSET

0,2

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

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

FORMULA

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

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A014200, A057655.

Sequence in context: A299771 A375027 A294963 * A316316 A333288 A159786

Adjacent sequences: A014195 A014196 A014197 * A014199 A014200 A014201

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved