A071383 -id:A071383 - cp's OEIS frontend

A071384 Radii of the circles around (0,0) that contain record numbers of lattice points, rounded up to the next integer.

0, 1, 3, 5, 9, 19, 34, 65, 75, 167, 269, 372, 401, 896, 1444, 2002, 2435, 5445, 8779, 12175, 15591, 34862, 56213, 77953, 113501, 231769, 253794, 409231, 567501, 886464, 1687299, 1982193, 3196190, 4432317, 7146896, 13178226, 15980946

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

a(n)^2 = A071383(n) for a(n) = 1, 5, 65, ... .

Ray Chandler, Table of n, a(n) for n = 1..425
Hugo Pfoertner, Construction of A071383, A071384, A071385

Cf. A071383, A071385, A046109.

a(n) = ceiling ( A071383(n)^(1/2) ).

Description clarified by Günter Rote, Sep 13 2023

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

A326312 Where the number of divisors d(k) reaches a new record for numbers k whose prime factors are of the form 3*j+2.

Original entry on oeis.org

2, 4, 8, 16, 20, 40, 80, 160, 320, 400, 440, 800, 880, 1600, 1760, 3520, 4400, 7040, 8800, 14960, 17600, 29920, 59840, 74800, 119680, 149600, 299200, 598400, 1196800, 1376320, 1720400, 2752640, 3440800, 6881600, 13763200, 27526400, 34408000, 49891600, 68816000

Offset: 1

Hugo Pfoertner, Sep 11 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300
Amiram Eldar, A230655 & A071383 and (3, 1)- and (4, 1)-highly composite numbers, thread in SeqFan mailing list, Sep 11 2019.

Cf. A053624, A071383, A230655, A326313, A326314.

aQ[n_] := AllTrue[FactorInteger[n][[;; , 1]], Mod[#, 3] == 2 &]; s[n_] := DivisorSum[n, 1 &, aQ[#] &]; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 2, 10^5}]; seq (* Amiram Eldar, Sep 12 2019 *)

pkn(x,d,m)={my(fn=factor(x),nf=#fn[,1]);for(k=1,nf,if(fn[k,1]%d!=m,return(0))); numdiv(x)};
divrecord=0;
for(k=2,50000000,my(j=pkn(k,3,2));if(j>divrecord,divrecord=j;print1(k,", ")))

More terms from Amiram Eldar, Sep 12 2019

A326313 Where the number of divisors d(k) reaches a new record for numbers k whose prime factors are of the form 4*j+3.

Original entry on oeis.org

3, 9, 21, 63, 189, 441, 567, 693, 2079, 4851, 6237, 13167, 39501, 92169, 118503, 276507, 829521, 908523, 2119887, 2725569, 6359661, 19078983, 28164213, 57236949, 84492639, 171710847, 197149491, 591448473, 1211061159, 1774345419, 3633183477, 5323036257, 6505933203

Offset: 1

Hugo Pfoertner, Sep 11 2019

nonn

For references and links see A326312.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A053624, A071383, A230655, A326312, A326314.

aQ[n_] := AllTrue[FactorInteger[n][[;; , 1]], Mod[#, 4] == 3 &]; s[n_] := DivisorSum[n, 1 &, aQ[#] &]; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 2, 10^5}]; seq (* Amiram Eldar, Sep 12 2019 *)

pkn(x,d,m)={my(fn=factor(x),nf=#fn[,1]);for(k=1,nf,if(fn[k,1]%d!=m,return(0)));numdiv(x)};
divrecord=0;
for(k=2,50000000,my(j=pkn(k,4,3));if(j>divrecord,divrecord=j;print1(k,", ")))

More terms from Amiram Eldar, Sep 12 2019

A326314 Where the number of divisors d(k) reaches a new record for numbers k whose prime factors are of the form 6*j+5.

Original entry on oeis.org

5, 25, 55, 275, 935, 3025, 4675, 21505, 51425, 107525, 537625, 1182775, 2688125, 3118225, 15591125, 34300475, 77955625, 127847225, 583108075, 639236125, 1406319475, 3196180625, 6008819575, 23907431075, 30044097875, 66097015325, 150220489375, 318467437475, 1123649260525

Offset: 1

Hugo Pfoertner, Sep 11 2019

nonn

For references and links see A326312.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A053624, A071383, A230655, A326312, A326313.

aQ[n_] := AllTrue[FactorInteger[n][[;; , 1]], Mod[#, 6] == 5 &]; s[n_] := DivisorSum[n, 1 &, aQ[#] &]; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 2, 10^5}]; seq (* Amiram Eldar, Sep 12 2019 *)

pkn(x, d, m)={my(fn=factor(x), nf=#fn[, 1]); for(k=1, nf, if(fn[k, 1]%d!=m, return(0))); numdiv(x)};
divrecord=0;
for(k=2, 200000000, my(j=pkn(k, 6, 5)); if(j>divrecord, divrecord=j; print1(k, ", ")))

More terms from Amiram Eldar, Sep 12 2019

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

Jianing Song, May 20 2021

nonn

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

			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.

Jianing Song, Table of n, a(n) for n = 1..424

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).

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

a(n) = A071385(n+1)/4.

a(n) = A000005(A071383(n+1)) = A002654(A071383(n+1)).

A326311 Least numbers k such that A004018(k) is nondecreasing.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 13, 17, 20, 25, 50, 65, 85, 125, 130, 145, 170, 185, 205, 221, 250, 260, 265, 290, 305, 325, 425, 650, 725, 845, 850, 925, 1025, 1105, 1625, 1885, 2125, 2210, 2405, 2465, 2665, 3145, 3250, 3445, 3485, 3625, 3770, 3965, 4225, 5525

Offset: 1

Hugo Pfoertner, Sep 11 2019

nonn

Least squared radius of a circle around a grid point of the square lattice such that the number of grid points on this circle is not smaller than the number of grid points on any circle around a grid point of the square lattice with smaller radius. a(1) = 0 by convention.

Hugo Pfoertner, Table of n, a(n) for n = 1..440

Cf. A004018, A071383, A071385, A326421.

using Nemo
function A326311List(len)
    R, x = PolynomialRing(ZZ, "x")
    e = theta_qexp(2, len, x)
    L = [coeff(e, j) for j in 0:len - 1]
    m = ZZ(0)
    [n - 1 for (n, l) in enumerate(L) if l == (m = max(m, l))]
end
A326311List(1000) |> println # Peter Luschny, Sep 12 2019

r2=0;for(k=0,6000,my(a004018 = if( k<1, k==0, 4 * sumdiv( k, d, (d%4==1) - (d%4==3))));if(a004018>=r2,r2=a004018;print1(k,", ")))

cp's OEIS Frontend

A071384 Radii of the circles around (0,0) that contain record numbers of lattice points, rounded up to the next integer.

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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.

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A326312 Where the number of divisors d(k) reaches a new record for numbers k whose prime factors are of the form 3*j+2.

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A326313 Where the number of divisors d(k) reaches a new record for numbers k whose prime factors are of the form 4*j+3.

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A326314 Where the number of divisors d(k) reaches a new record for numbers k whose prime factors are of the form 6*j+5.

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Mathematica

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A344470 Record values in A002654.

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A326311 Least numbers k such that A004018(k) is nondecreasing.

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