A216467 - OEIS

%I #27 Sep 02 2019 14:40:45

%S 21,35,39,45,51,55,57,69,75,77,85,87,91,93,95,99,105,111,115,117,119,

%T 123,133,135,141,143,145,147,153,155,159,161,165,171,175,177,183,185,

%U 187,189,195,201,203,205,207,209,213,215,217,219,221,225,231,235,237,244,245

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

%C See A178793, A178794 for terminology.

%C 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

%C From _Charlie Neder_, Jun 27 2018: (Start)

%C 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:

%C a) k is odd:

%C   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.

%C   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.

%C b) k is even:

%C   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)

%C   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.

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

%t Select[Range[300], If[OddQ[#], !PrimePowerQ[#] && !PrimePowerQ[# - 1] && !PrimePowerQ[# + 1], PrimeOmega[# - 1] > 2 && PrimeOmega[# + 1] > 2]&] (* _Jean-François Alcover_, Sep 02 2019, after _Andrew Howroyd_ *)

%o (PARI) select(k->if(k%2, !isprimepower(k) && !isprimepower(k-1) && !isprimepower(k+1), omega(k-1)>2 && omega(k+1)>2), [1..300]) \\ _Andrew Howroyd_, Jun 27 2018

%Y Cf. A178793, A178794, A157428, A157429.

%K nonn

%O 1,1

%A _Gregg Whisler_, Sep 07 2012

%E Several missing terms added by _Charlie Neder_, Jun 27 2018

%E More terms from _Jean-François Alcover_, Sep 02 2019