cp's OEIS Frontend

From Gregg Whisler, Jun 21 2010: (Start)

a(n) is also A157428 + 1. [Charles R Greathouse IV points out that this is false, since (1308, 1274) is in (A157428, A157429) but not in (A178793, A178794). Oct 17 2012]

An isolated lattice point is surrounded (in a Moore neighborhood of r=1) in the Z^2 lattice of points by 8 points that are not visible from the origin. (End)

An isolated lattice point is surrounded (in a Moore neighborhood) by 8 points that are not visible from the origin. I have also submitted the corresponding sequence of denominators.

From Gregg Whisler, Jun 21 2010: (Start)

a(n) is also A157429 + 1.

These are also the x coordinates of the isolated visible lattice points in Z^2. (End)

See A178793, A178794 for terminology.

It is not clear to me how many - if any! - of these terms are known to be correct. - N. J. A. Sloane, Oct 17 2012

From Charlie Neder, Jun 27 2018: (Start)

For row k to contain an isolated lattice point, k must contain a pair (m-1,m+1) of nontotatives, and both k-1 and k+1 must contain a triple of consecutive nontotatives. The CRT can then be used to "align" the groups into a box containing a lattice point. We consider the cases when k is odd and when k is even:

a) k is odd:

k cannot be a prime p or a power of a prime, because then the nontotatives to k are precisely the multiples of p, which contain no pairs since k is odd and therefore p > 2. As long as k is divisible by at least two odd primes, a pair can be found by the CRT.

k-1 and k+1 are even but cannot be powers of two, since then the nontotatives would be the even numbers, which contain no triples. As long as they each have at least one odd divisor, then all the odd nontotatives will be centers of triples.

b) k is even:

There are no other restrictions on k itself, since pairs are very easy to find for even k. (e.g. for any prime p not dividing k, (p-1,p+1) is a valid pair)

k-1 and k+1 are both odd and must be the products of at least three distinct primes, since a triple could not form otherwise. The CRT can be used to find triples as long as this is the case.

The first such even k is 664, with isolated point (189449,664) on it. (End)

Any grid point with relative coordinates (x,y) from the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 will have another point directly between it and the central point and will thus not be visible. In an infinite 2D square lattice the ratio of visible grid points to all points is 6/Pi^2, approximately 0.608, the same as the probability of two random numbers being relative prime.

For a square spiral of size 10001 X 10001, slightly over 100 million numbers, a total of 60803664 numbers are visible, of which 2155170 are prime. The total number of primes in the same range is 5762536, giving a ratio of visible primes to all primes of about 0.374. This is significantly lower than the ratio for all numbers of 0.608, indicating a prime is more likely to be hidden from the origin than a random number.

Primes p such that A174344(p) and A268038(p) are coprime. - Robert Israel, Feb 16 2024

Any grid point labeled with a prime number and with coordinates (x,y) relative to the central grid point, which is numbered 1, and where the greatest common divisor (gcd) of |x| and |y| equals 1 will be visible from the central point. Grid points where gcd(|x|,|y|) > 1 may have another prime grid point directly between it and the central point and will thus not be visible.

For a square spiral of size 10001 by 10001, slightly over 100 million numbers, a total of 5762536 primes are present, of which 4811013 are visible. This gives a ratio of visible primes to all primes of about 0.835.