A267309 Number of discrete vectors with integral components and integral length <= n in a 3-dimensional vector space (Partial sums of A016725).

6, 12, 42, 48, 78, 108, 162, 168, 270, 300, 378, 408, 486, 540, 690, 696, 798, 900, 1026, 1056, 1326, 1404, 1554, 1584, 1734, 1812, 2130, 2184, 2358, 2508, 2706, 2712, 3102, 3204, 3474, 3576, 3798, 3924, 4314, 4344, 4590, 4860, 5130, 5208, 5718

Offset

1,1

Comments

This sequence is Z_3(n), where Z_D(n) counts all vectors with integral components and length in a D-dimensional vector space within a certain radius. This sequence represents partial sums of A016725.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..150 from Christopher Heiling)

Formula

a(n) = Sum_{k=1..n} A005875(k^2).

a(n) = Sum_{k=1..n} A016725(k).

Example

For n = 2 the a(n)= 12 integral solutions of x^2 + y^2 + z^2 <= 2^2 are: {x,y,z} = {{0,0,1}; {0,1,0}; {1,0,0}; {0,0,-1}; {0,-1,0}; {-1,0,0}; {0,0,2}; {0,2,0}; {2,0,0}; {0,0,-2}; {0,-2,0}; {-2,0,0}}.

Prog

(Python)
from math import prod
from sympy import factorint
def A267309(n): return 6*sum(prod(p**e+(0 if p&3==1 else (p**e-1)//(p-1)<<1) for p, e in factorint(m>>(~m & m-1).bit_length()).items()) for m in range(1, n+1)) # Chai Wah Wu, Feb 22 2026

Crossrefs

Cf. A005875, A016725.

Sequence in context: A371234 A371235 A152786 * A206039 A048069 A152787

Adjacent sequences: A267306 A267307 A267308 * A267310 A267311 A267312

Keyword

nonn

Author

Christopher Heiling, Jan 19 2016

Status

approved