A230655 Squared radii of circles around a point of the hexagonal lattice that contain a record number of lattice points.

0, 1, 7, 49, 91, 637, 1729, 8281, 12103, 53599, 157339, 375193, 1983163, 4877509, 13882141, 85276009, 180467833, 596932063, 3428888827, 4178524441, 7760116819, 29249671087, 36412855843, 147442219561, 254889990901, 473367125959, 1784229936307, 2439661341481

Offset

1,3

Comments

It appears that this is also the sequence of numbers with a record number of divisors all of whose prime factors are of the form 3k + 1. - Amiram Eldar, Sep 12 2019 [This is correct, see A343771. - Jianing Song, May 19 2021]

Indices of records of A004016. Apart from the first term, also indices of records of A002324. - Jianing Song, May 20 2021

Links

Jianing Song, Table of n, a(n) for n = 1..247 (all terms <= 10^75.)

Example

a(2)=7 because a circle with radius sqrt(7) around the lattice point at (0,0) is the first circle that passes through more lattice points than a circle with radius 1, which passes through 6 points. The 12 hit points are (+-1/2,+-3*sqrt(3)/2), (+-2,+-sqrt(3)), (+-5/2, +-sqrt(3)/2).

Prog

(PARI) my(v=list_A344473(10^15), rec=0); print1(0, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(v[n], ", "))) \\ Jianing Song, May 20 2021, see program for A344473

Crossrefs

Cf. A004016, A002324.

Cf. A003136 (all occurring squared radii), A198799 (common terms), A230656 (index positions of records), A344472 (records).

Apart from the first term, subsequence of A343771.

Indices of records of Sum_{d|n} kronecker(m, d): this sequence (m=-3), A071383 (m=-4, similar sequence for square lattice), A279541 (m=-6).

Sequence in context: A161145 A343737 A344473 * A357302 A343771 A008461

Adjacent sequences: A230652 A230653 A230654 * A230656 A230657 A230658

Keyword

nonn

Author

Hugo Pfoertner, Oct 27 2013

Extensions

Offset corrected by Jianing Song, May 20 2021

Status

approved