A230656 -id:A230656 - cp's OEIS frontend

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

Hugo Pfoertner, Oct 27 2013

nonn

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

			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).

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

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).

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

Offset corrected by Jianing Song, May 20 2021

A075880 Position of the circles around (0,0) that contain record numbers of lattice points in the list of all circles around (0,0) that pass through lattice points, ordered by increasing radius.

Original entry on oeis.org

0, 1, 4, 13, 30, 121, 362, 1232, 1584, 7121, 17548, 32649, 37603, 174926, 437750, 821432, 1198677, 5678338, 14335447, 27044791, 43735981, 209473053, 531787054, 1006745669, 2097411347, 8474384496, 10122355701

Offset: 1

Hugo Pfoertner, Oct 16 2002

nonn

The first terms of this sequence were given by James Buddenhagen in a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points"

			a(4)=13 because A001481(14) = A071383(4)=25.

James Buddenhagen, Circle with 3 lattice points, thread in NG sci.math.
Hugo Pfoertner, Asymptotic Behavior of A075880(n)

Cf. A001481, A064533, A071383, A230656.

a(n) = k-1 for which A001481(k) = A071383(n).

lim n ->infinity a(n) = k_LR * exp(n) / n^(1/2), where k_LR is the Landau-Ramanujan constant 0.764223653... (A064533)

Minor edits to adjust formula and example for changes to offset of related sequences by Ray Chandler, Jan 13 2012

cp's OEIS Frontend

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

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