A059743 Number of points of norm^2 <= n^2 in the square lattice that are visible from the origin.

1, 5, 9, 17, 33, 49, 73, 89, 121, 153, 193, 225, 265, 313, 385, 441, 481, 545, 617, 673, 769, 833, 929, 1001, 1113, 1193, 1281, 1385, 1489, 1585, 1705, 1817, 1961, 2073, 2225, 2345, 2481, 2601, 2753, 2913, 3065, 3185, 3361, 3481, 3697, 3857, 4017, 4177

Offset

0,2

Comments

A lattice point (i,j) is "visible" from the origin if no other lattice point lies on the line segment from (0,0) to (i,j), which is equivalent to saying that i and j are relatively prime. By convention, we say that (0,0) is visible from the origin.

References

Tom M. Apostol, "Introduction to Analytic Number Theory", Springer-Verlag, Section 3.8.

Links

Robert Israel, Table of n, a(n) for n = 0..7152

Maple

N:= 100; # to get a(0) to a(N)
for n from 1 to N do
  C[n]:= Array(1..n, j -> `if`(igcd(n, j)=1, 1, 0)):
  B[n]:= map(round, Statistics[CumulativeSum](C[n]));
od:
1, 5, 9, seq(1 + 8*(add(numtheory[phi](x), x=1..floor(n/sqrt(2)))+add(B[x][floor(sqrt(n^2-x^2))], x=ceil(n/sqrt(2))..n-1)), n = 3 .. N); # Robert Israel, Jun 15 2014

Crossrefs

See also A000328, where visibility is not required.

Sequence in context: A160426 A301786 A258411 * A000322 A205539 A020737

Adjacent sequences: A059740 A059741 A059742 * A059744 A059745 A059746

Keyword

nonn

Author

John W. Layman, Oct 13 2001

Status

approved