A230655 -id:A230655 - cp's OEIS frontend

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

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

Jianing Song, May 20 2021

nonn

Record values of A004016.

			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.

Jianing Song, Table of n, a(n) for n = 1..247

Cf. A230655, A004016, A002324, A344472, A000005, A344473.

Cf. A071385 (similar sequence).

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

a(n) = A004016(A230655(n)).

a(n) = 6*A000005(A230655(n)) = 6*A002324(A230655(n)), n > 1.

A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

Original entry on oeis.org

0, 1, 5, 25, 65, 325, 1105, 4225, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

The number of lattice points (i,j) on the circle with i^2 + j^2 = a(n) is given by A071385(n).
In a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points", James R. Buddenhagen asked: Which circles have the property that they pass through more lattice points than any smaller circle? and he gave the terms 1, 25, 65, 325, 1105, 4225, 5525, with the missing 5 added by Ahmed Fares. In the same thread Gerry Myerson mentioned the factorization into primes of the form 4*k+1.
Also, numbers with a record number of divisors all of whose prime factors are of the form 4k + 1. - Amiram Eldar, Sep 12 2019
Indices of records of A004018. Apart from the first term, also indices of records of A002654. - Jianing Song, May 20 2021

Ray Chandler, Table of n, a(n) for n = 1..425 (first 97 terms from Ray Chandler, terms 98-365 from Amiram Eldar)
James Buddenhagen, Circle with 3 lattice points, thread in sci.math (May 2002)
Hugo Pfoertner, Construction of the sequences A071383, A071384, A071385

Cf. A000448, A048610, A052199, A071384, A071385, A230655, A300162. Subsequence of A054994 (excluding first term). Where records occur in A004018. See A088959 for circles with integer radius.

Indices of records of Sum_{d|n} kronecker(m, d): A230655 (m=-3), this sequence (m=-4), A279541 (m=-6).

my(v=list(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 A054994

from math import prod
from sympy import isprime
primes_congruent_1_mod_4 = [5]
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
def generate_A054994():
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(r,exponents):
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0:
        yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
n,record,radius=1,1,0
print(radius, end="") # or record, for A071385
for radius,expo in generate_A054994():
    num_points = 4*prod((e+1) for e in expo)
    if num_points>record:
        record = num_points
        n += 1
        print (",", radius, end="") # or record, for A071385
        if n==27: break
print()
# Günter Rote, Sep 12 2023

For n>1 we have 1 < a(n+1)/a(n) <= 5, since one can multiply the points x+iy for which x^2 + y^2 = N by either 2+i or 2-i to get two new sets of points X+iY for which X^2 + Y^2 = 5N. This strictly increases the number since it is easy to see that the two sets aren't the same. - J. H. Conway, Jun 04 2002

lim n ->infinity Log(a(n))/n = 1. [Conjectured by Benoit Cloitre, proved by J. H. Conway]

A343771 Smallest k such that circle centered at the origin and with radius sqrt(k) passes through exactly 6*n integer points in the hexagonal lattice (see A004016).

Original entry on oeis.org

1, 7, 49, 91, 2401, 637, 117649, 1729, 8281, 31213, 282475249, 12103, 13841287201, 1529437, 405769, 53599, 33232930569601, 157339, 1628413597910449, 593047, 19882681, 3672178237, 3909821048582988049, 375193, 68574961, 179936733613, 2989441, 29059303, 459986536544739960976801, 7709611

Offset: 1

Jianing Song, Apr 28 2021

nonn

a(n) is the smallest k such that A004016(k) = 6*n.
Also a(n) is the smallest index of n in A002324.
a(n) is the smallest term in A004611 that has exactly n divisors.

			91 = 7 * 13 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 4 divisors, so a(4) = 91.
8281 = 7^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 3 with exactly 9 divisors, so a(9) = 8281.

Jianing Song, Table of n, a(n) for n = 1..1000
Helmut Ruhland, A family of lattices with an unbounded number of unit vectors, arXiv:2410.16172 [math.MG], 2024. See p. 2.

Cf. A004016, A002324, A004611, A230655, A355919.

Analog of A005179 and A018782.

primelist(d, r, l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p, d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A343771(n)=my(pf=prodR(n, n), a=1, b, v=primelist(3, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bA005179.

If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 3. The smallest n such that the equality does not hold is n = 128. [Those n such that the equality does not hold are listed in A355919. - Jianing Song, Jul 21 2022]
a(n) <= 7^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 7^(p-1) * 13^(q-1) for primes p >= q.

A344472 Record values in A002324.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 128, 144, 160, 192, 216, 256, 288, 320, 384, 432, 512, 576, 640, 768, 864, 1024, 1152, 1280, 1536, 1728, 2048, 2304, 2560, 3072, 3456, 3840, 4096, 4608, 5120, 5184, 6144, 6912, 7680, 8192, 9216

Offset: 1

Jianing Song, May 20 2021

nonn

Also numbers k such that A343771(m) > A343771(k) for all m > k.

			8 is a term because the circle with radius sqrt(1729) centered at the origin hits exactly 6*8 = 48 points in the A_2 lattice, and any circle with radius < sqrt(1729) centered at the origin hits fewer than 48 points.

Jianing Song, Table of n, a(n) for n = 1..246

Cf. A230655, A344471, A002324, A343771, A000005, A344473.

Records of Sum_{d|n} kronecker(m, d): this sequence (m=-3), A344470 (m=-4), A279542 (m=-6).

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

a(n) = A344471(n+1)/6.

a(n) = A000005(A230655(n+1)) = A002324(A230655(n+1)).

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

A230656 Position of the circles with center at a lattice point that contain record numbers of lattice points, in the list of Loeschian numbers A003136.

Original entry on oeis.org

1, 2, 5, 21, 34, 186, 454, 1927, 2746, 11128, 30906, 70762, 349068, 830170, 2278644, 13219495, 27373350, 87615127

Offset: 0

Hugo Pfoertner, Oct 27 2013

			a(3) = 21 because A230655(3) = A003136(21) = 49.

Cf. A003136 (Loeschian numbers), A230655, A075880 (similar sequence for square lattice)

a(n) = k for which A230655(n) = A003136(k).

A326421 Least numbers k such that A088534(k) is nondecreasing.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 91, 133, 147, 169, 196, 217, 247, 259, 273, 301, 343, 361, 364, 399, 403, 427, 441, 469, 481, 507, 511, 532, 553, 559, 588, 589, 637, 931, 1183, 1519, 1729, 2821, 3367

Offset: 1

Hugo Pfoertner, Sep 12 2019

nonn

Least squared radius of a circle around a grid point of the hexagonal 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 hexagonal lattice with smaller radius. a(1) = 0 by convention.

Hugo Pfoertner, Table of n, a(n) for n = 1..163

Cf. A003136, A088534, A118886, A230655, A326311.

\\ after M. F. Hasler's A088534(n,d)
r2=0;for(k=0,3500,my(a088534=sum(x=0, sqrt(k\3), sum(y=max(x, sqrtint(k-x^2)\2), sqrtint(k-2*x^2), x^2+x*y+y^2==k)));if(a088534>=r2,r2=a088534;print1(k,", ")))

A357302 Numbers k such that k^2 can be represented as x^2 + x*y + y^2 in more ways than for any smaller k.

Original entry on oeis.org

1, 7, 49, 91, 637, 1729, 12103, 53599, 375193, 1983163, 13882141, 85276009, 596932063, 4178524441, 5201836549, 36412855843, 254889990901, 348523048783, 2439661341481, 17077629390367, 25442182561159, 178095277928113, 1246666945496791, 2009932422331561, 14069526956320927

Offset: 1

Hugo Pfoertner, Sep 25 2022

nonn

Apparently the number of grid points t(n) = {1, 2, 3, 5, 8, 14, 23, ...} (A357303) in the reduced representations as described in the examples matches t(n) = A087503(n-3) + 2 for n >= 3, i.e., t(n) = t(n-1) + 3*t(n-2) - 3*t(n-3) for n >= 5. This coincidence persists up to t(15) = 1823, but t(16) = 2553, whereas the recurrence predicts 3281, which is t(17). It seems that all of the terms generated by the recurrence also appear as record numbers of grid points. However, there are other record numbers in between, of which 2553 is the first occurrence.

			The essential information in the complete set of representations of a square a(n)^2 can be extracted by taking into account the symmetries of the triangular lattice. If r is the number of all representations of a(n)^2, then there are t = (r/6 + 1)/2 pairs of triangular oblique coordinates lying in a sector of angular width Pi/6 completely containing the essential information.
a(1) = 1: r = 6 representations of 1^2 are [-1, 0], [-1, 1], [0, -1], [0, 1], [1, -1], [1, 0] reduced: (6/6 + 1)/2 = 1 grid point [1,0].
a(2) = 7: r = 18 representations of 7^2 = 49 are [-8, 5], [-7, 0], [-7, 7], [-5, -3], [-5, 8], [-3, -5], [-3, 8], [0, -7], [0, 7], [3, -8], [3, 5], [5, -8], [5, 3], [7, -7], [7, 0], [8, -5], [8, -3], [8, 3]; reduced: (18/6 + 1)/2 = 2 grid points [7, 0], [8, 3].
After a(2) = 7 there are no squares with more than 18 representations, e.g., r = 18 for 13^2, 14^2, 19^2, 21^2, ..., 42^2, 43^2.
a(3) = 49: r = 30 representations of 49^2 = 2401 are [-56, 21], [-56, 35], [-55, 16], [-55, 39], [-49, 0], [-49, 49], [-39, -16], [-39, 55], [-35, -21], [-35, 56], [-21, -35], [-21, 56], [-16, -39], [-16, 55], [0, -49], [0, 49], [16, -55], [16, 39], [21, -56], [21, 35], [35, -56], [35, 21], [39, -55], [39, 16], [49, -49], [49, 0], [55, -39], [55, -16], [56, -35], [56, -21]; reduced: (30/6 + 1)/2 = 3 grid points [49, 0], [55, 16], [56, 21].
There are no squares with r > 18 between 49 and 90.
a(4) = 91: r = 54 representations of 91^2 = 8281 are [-105,49], [-105,56], ..., [105, -56], [105,-49]; reduced: (54/6 + 1)/2 = 5 grid points [91, 0], [96, 11], [99, 19], [104, 39], [105, 49].

Cf. A002324, A003136, A004016, A050931, A088534, A230655, A357303.

Cf. A087503, A246360.

a357302(upto) = {my (dmax=0);for (k = 1, upto, my (d = #qfbsolve (Qfb(1,1,1), k^2, 3)); if(d > dmax, print1(k,", "); dmax=d))};
a357302(400000)

\\ more efficient using function list_A344473 (see there)
a355703(maxexp10)= {my (sqterms=select(x->issquare(x), list_A344473 (10^(2*maxexp10))), r=0); for (k=1, #sqterms, my (d = #qfbsolve(Qfb(1,1,1),v[k],3)); if (d>r, print1(sqrtint(v[k]),", "); r=d))};
a355703(17)

from itertools import count, islice
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A357302_gen(): # generator of terms
    c = 0
    for k in count(1):
        if (d:=len(diop_quadratic(x*(x+y)+y**2-k**2))) > c:
            yield k
            c = d
A357302_list = print(list(islice(A357302_gen(),6))) # Chai Wah Wu, Sep 26 2022

cp's OEIS Frontend

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

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A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

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A343771 Smallest k such that circle centered at the origin and with radius sqrt(k) passes through exactly 6*n integer points in the hexagonal lattice (see A004016).

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A344472 Record values in A002324.

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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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A230656 Position of the circles with center at a lattice point that contain record numbers of lattice points, in the list of Loeschian numbers A003136.

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