A090025 -id:A090025 - cp's OEIS frontend

A081474 Number of distinct lines through the origin in n-dimensional cube of side length n.

0, 1, 5, 49, 529, 7471, 112825, 2078455, 42649281, 997784221, 25875851825, 742641202183, 23283999690561, 793616663524231, 29188521870580929, 1152885848976064513, 48659336030073207425, 2185894865613157551481, 104126348669497256201905, 5242869988601103651841105

Offset: 0

Joshua Zucker, Nov 25 2003

nonn

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

			a(3) = 49 because in the 3-dimensional lattice of side length 3, the lines through the origin are determined by all 37 points with at least one coordinate = 3 and 6 permutations of (2,1,0) and 3 permutations each of (2,1,1) and (2,2,1).

Alois P. Heinz, Table of n, a(n) for n = 0..386

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.

a:= n-> add(numtheory[mobius](i)*((floor(n/i)+1)^n-1), i=1..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2022

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[k, k], {k, 0, 20}]

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

A339756 Mark each point on the n X n X n grid with the number of points that are visible from it; a(n) is the number of distinct values in the grid.

Original entry on oeis.org

1, 4, 4, 8, 4, 17, 12, 15, 14, 33, 12, 58, 28, 43, 52, 113, 39, 140, 57, 124, 129, 240, 66, 241, 173, 270, 217, 362, 58, 388, 292, 454, 351, 539, 166, 783, 471, 723, 463, 880, 229, 1134, 642, 843, 763, 1441, 311, 1415, 740, 1295, 987, 1888, 357, 1629, 1063, 1750, 1231, 2381, 289, 2652

Offset: 1

Torlach Rush, Dec 15 2020

nonn

a(n) <= A058187(n). This is because A058187(n) is the maximum number of points required to calculate a(n).

			a(1) = 1 because there are 7 visible points from every point on the grid.
a(2) = 4 because 19 points are visible from every vertex of the grid, 23 points are visible from the midpoint of every edge of the grid, 25 points are visible from the midpoint of every face of the grid, and 26 points are visible from the middle of the grid.
a(3) = 4 because 49 points are visible from every vertex of the grid, 53 points are visible from the inner points of every edge of the grid, 55 points are visible from the inner points of every face of the grid, and 56 points are visible from the inner points of the grid.

Eric Weisstein's World of Mathematics, Visible Point

Cf. A049687, A049691, A058187, A090025, A339400.

\\ n = side length, d = dimension
cdvps(n, d) ={my(m=Map());
  forvec(u=vector(d, i, [0, n\2]),
    my(c=0); forvec(v=[[t-n, t]|t<-u], c+=(gcd(v)==1));
    mapput(m, c, 1), 1);
  #m; }
a(n) = cdvps(n, 3)

A175549 Number of triples (a, b, c) with gcd(a, b, c) = 1 and -n <= a,b,c <= n.

Original entry on oeis.org

0, 26, 98, 290, 578, 1154, 1730, 2882, 4034, 5762, 7490, 10370, 12674, 16706, 20162, 24770, 29378, 36290, 41474, 50114, 57026, 66242, 74882, 87554, 96770, 111170, 123266, 138818, 152642, 172802, 186626, 209666, 228098, 251138, 271874, 299522

Offset: 0

Charles R Greathouse IV, Jun 24 2010

nonn

Indranil Ghosh, Table of n, a(n) for n = 0..1000

Cf. A090025, A018805, A049691.

Table[If[n>0, 8 * Sum[MoebiusMu[k] * ((Floor[n/k] + 1)^3 - 1), {k, 1, n}] - 24 * Sum[EulerPhi[k], {k, 1, n}] - 6, 0], {n, 0, 35}] (* Indranil Ghosh, Mar 11 2017 *)

a(n)=if(n>0,8*sum(k=1,n,moebius(k)*((n\k+1)^3-1))-24*sum(k=1,n,eulerphi(k))-6)

from functools import lru_cache
@lru_cache(maxsize=None)
def A175549(n):
    if n == 0:
        return 0
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A175549(k1)
        j, k1 = j2, n//j2
    return 4*n*(n - 1)*(2*n + 5)-c+26*(j-1)# Chai Wah Wu, Mar 30 2021

For n > 0, a(n) = 8*A090025(n) - 12*A018805(n) - 18.

a(n) = 2*n*(4*n^2 + 6*n + 3) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

Edited by Charles R Greathouse IV, Jul 19 2010

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A081474 Number of distinct lines through the origin in n-dimensional cube of side length n.

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A339756 Mark each point on the n X n X n grid with the number of points that are visible from it; a(n) is the number of distinct values in the grid.

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A175549 Number of triples (a, b, c) with gcd(a, b, c) = 1 and -n <= a,b,c <= n.

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