A014200 Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

0, 1, 2, 2, 3, 5, 5, 5, 6, 7, 9, 9, 9, 11, 11, 11, 12, 14, 15, 15, 17, 17, 17, 17, 17, 20, 22, 22, 22, 24, 24, 24, 25, 25, 27, 27, 28, 30, 30, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 37, 40, 40, 42, 44, 44, 44, 44, 44, 46

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From Ant King, Mar 15 2013: (Start)

The terms of this sequence give a running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors.

To see how good the approximation n * Pi/4 is to a(n), note that a(10^6) = 785387 whereas 10^6 * Pi/4 rounds to 785398. (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A014198, A059851, A101455.

Partial sums of A002654.

Programs

Mathematica

1/4*Prepend[SquaresR[2,#]&/@Range[58],0]//Accumulate (* Ant King, Mar 15 2013 *)

PARI

a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d))); \\ Seiichi Manyama, Dec 18 2021

Formula

a(n) = A014198(n) / 4.

Limit_{n->infinity} a(n)/n = Pi/4 = A003881.

a(n) = n - floor(n/3) + floor(n/5) - floor(n/7) + floor(n/9) - floor(n/11) + ... - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Dec 23 2016