A090028 Number of distinct lines through the origin in 6-dimensional cube of side length n.

0, 63, 665, 3969, 14833, 45801, 112825, 257257, 515025, 980217, 1720145, 2934505, 4693473, 7396137, 11112129, 16464385, 23555441, 33430033, 45927505, 62881561, 83865257, 111331241, 144772201, 187839225, 238778281, 303522401, 379323785

Offset

0,2

Comments

Equivalently, lattice points where the GCD of all coordinates = 1.

Links

Table of n, a(n) for n=0..26.

Formula

a(n) = A090030(6, n).

a(n) = (n+1)^6 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

Example

a(2) = 665 because the 665 points with at least one coordinate=2 all make distinct lines and the remaining 63 points and the origin are on those lines.

Mathematica

aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]]; lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1; Table[lines[6, k], {k, 0, 40}]

Prog

(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A090028(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090028(k1)
        j, k1 = j2, n//j2
    return (n+1)**6-c+63*(j-n-1) # Chai Wah Wu, Mar 30 2021

Crossrefs

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.

Sequence in context: A228221 A022522 A152731 * A152725 A086578 A198399

Adjacent sequences: A090025 A090026 A090027 * A090029 A090030 A090031

Keyword

nonn

Author

Joshua Zucker, Nov 25 2003

Status

approved