A090027 Number of distinct lines through the origin in 5-dimensional cube of side length n.

0, 31, 211, 961, 2851, 7471, 15541, 31471, 55651, 95821, 152041, 239791, 351331, 517831, 723241, 1007041, 1352041, 1821721, 2359051, 3082921, 3904081, 4956901, 6151651, 7677901, 9334261, 11445361, 13746181, 16566691, 19644031, 23432851, 27408331, 32333581

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..31.

Formula

a(n) = A090030(5, n).

a(n) = (n+1)^5 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

Example

a(2) = 211 because the 211 points with at least one coordinate=2 all make distinct lines and the remaining 31 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[5, k], {k, 0, 40}]

Prog

(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A090027(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090027(k1)
        j, k1 = j2, n//j2
    return (n+1)**5-c+31*(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: A022521 A152730 A361700 * A164784 A290008 A121616

Adjacent sequences: A090024 A090025 A090026 * A090028 A090029 A090030

Keyword

nonn

Author

Joshua Zucker, Nov 25 2003

Status

approved