A073092 Number of numbers of the form x^2 + y^2 (0 <= x <= y) less than or equal to n.

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 13, 13, 13, 13, 13, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 27, 27, 28, 29, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 34, 34, 34, 35, 35, 35, 35, 36, 37, 38

Offset

0,2

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

a(n) = Sum_{k=0..n} A000161(k).

a(n) is asymptotic to Pi*n/8.

Example

0^2 + 0^2, 0^2 + 1^2, 1^2 + 1^2 are less than or equal to 2 hence a(2) = 3.

Mathematica

Accumulate @ Table[Length @ PowersRepresentations[n, 2, 2], {n, 0, 100}] (* Amiram Eldar, Mar 08 2020 *)

Prog

(PARI) a(n)=sum(x=0, n, sum(y=0, x, if((sign(x^2+y^2-n)+1)*sign(x^2+y^2-n), 0, 1)))
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A073092_gen(): # generator of terms
    yield (c:=1)
    for n in count(1):
        f = factorint(n)
        c += int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1)
        yield c
A073092_list = list(islice(A073092_gen(), 30)) # Chai Wah Wu, Sep 08 2022

Crossrefs

Cf. A000161, A001481.

Cf. A057655.

Sequence in context: A165706 A242453 A241151 * A088023 A324477 A287292

Adjacent sequences: A073089 A073090 A073091 * A073093 A073094 A073095

Keyword

easy,nonn

Author

Benoit Cloitre, Aug 18 2002

Status

approved