cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, Oct 27 2013

Keywords

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

Examples

			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).

Links

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).

Programs

  • 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

Extensions

Offset corrected by Jianing Song, May 20 2021

A344470 Record values in A002654.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 144, 160, 192, 216, 256, 288, 320, 384, 432, 512, 576, 640, 768, 864, 960, 1024, 1152, 1280, 1536, 1728, 1920, 2048, 2304, 2560, 2880, 3072, 3456, 3840, 4096, 4608, 5120, 5760, 6144
Offset: 1

Views

Author

Jianing Song, May 20 2021

Keywords

Comments

Also numbers k such that A018782(m) > A018782(k) for all m > k.

Examples

			9 is a term because the circle with radius sqrt(4225) centered at the origin hits exactly 4*9 = 36 integer points, and any circle with radius < sqrt(4225) centered at the origin hits fewer than 36 points.

Links

Crossrefs

Cf. A071383, A071385, A002654, A018782, A000005, A054994.
Records of Sum_{d|n} kronecker(m, d): A344472 (m=-3), this sequence (m=-4), A279542 (m=-6).

Programs

  • PARI
    my(v=list(10^15), rec=0); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(rec, ", "))) \\ see program for A054994

Formula

a(n) = A071385(n+1)/4.
a(n) = A000005(A071383(n+1)) = A002654(A071383(n+1)).

A344471 Number of points in the hexagonal lattice A_2 on the circle centered at the origin with squared radius A230655(n).

Original entry on oeis.org

1, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 384, 432, 576, 648, 768, 864, 960, 1152, 1296, 1536, 1728, 1920, 2304, 2592, 3072, 3456, 3840, 4608, 5184, 6144, 6912, 7680, 9216, 10368, 12288, 13824, 15360, 18432, 20736, 23040, 24576, 27648
Offset: 1

Views

Author

Jianing Song, May 20 2021

Keywords

Comments

Record values of A004016.

Examples

			24 is a term because the circle with radius sqrt(91) centered at the origin hits exactly 24 points in the A_2 lattice, and any circle with radius < sqrt(91) centered at the origin hits fewer than 24 points.

Links

Crossrefs

Cf. A230655, A004016, A002324, A344472, A000005, A344473.
Cf. A071385 (similar sequence).

Programs

  • PARI
    my(v=list_A344473(10^15), rec=0); print1(1, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(6*rec, ", "))) \\ see program for A344473

Formula

a(n) = A004016(A230655(n)).
a(n) = 6*A000005(A230655(n)) = 6*A002324(A230655(n)), n > 1.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.