

A216467


Smallest numbers in the coordinates of the isolated visible lattice points in the infinite square grid.


2



21, 35, 39, 45, 51, 55, 57, 69, 75, 77, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 123, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205, 207, 209, 213, 215, 217, 219, 221, 225, 231, 235, 237, 244, 245
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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 (m1,m+1) of nontotatives, and both k1 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.
k1 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, (p1,p+1) is a valid pair)
k1 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)


LINKS

Table of n, a(n) for n=1..57.


MATHEMATICA

Select[Range[300], If[OddQ[#], !PrimePowerQ[#] && !PrimePowerQ[#  1] && !PrimePowerQ[# + 1], PrimeOmega[#  1] > 2 && PrimeOmega[# + 1] > 2]&] (* JeanFrançois Alcover, Sep 02 2019, after Andrew Howroyd *)


PROG

(PARI) select(k>if(k%2, !isprimepower(k) && !isprimepower(k1) && !isprimepower(k+1), omega(k1)>2 && omega(k+1)>2), [1..300]) \\ Andrew Howroyd, Jun 27 2018


CROSSREFS

Cf. A178793, A178794, A157428, A157429.
Sequence in context: A261408 A008946 A095738 * A330949 A248020 A290435
Adjacent sequences: A216464 A216465 A216466 * A216468 A216469 A216470


KEYWORD

nonn


AUTHOR

Gregg Whisler, Sep 07 2012


EXTENSIONS

Several missing terms added by Charlie Neder, Jun 27 2018
More terms from JeanFrançois Alcover, Sep 02 2019


STATUS

approved



