A300162 -id:A300162 - cp's OEIS frontend

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

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]

A299707 Numbers m such that m^2 + 1 can be expressed in more than one way as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

18, 32, 38, 43, 47, 57, 68, 70, 72, 73, 82, 83, 93, 98, 99, 107, 112, 117, 118, 122, 123, 128, 132, 133, 138, 142, 143, 148, 157, 162, 168, 172, 173, 174, 177, 182, 183, 187, 191, 192, 193, 200, 203, 207, 208, 212, 213, 216, 217, 218, 228, 232, 233, 237, 242, 243, 251, 252

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

			a(1) = 18: 18^2 + 1 = 325 = 17^2 + 6^2 = 15^2 + 10^2,
a(2) = 32: 32^2 + 1 = 1025 = 31^2 + 8^2 = 25^2 + 20^2,
a(5) = 47: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A050796, A299708, A300161, A300162, A300163, A300164, A300165, A300166.

A299708 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

325, 1025, 1445, 1850, 2210, 3250, 4625, 4901, 5185, 5330, 6725, 6890, 8650, 9605, 9802, 11450, 12545, 13690, 13925, 14885, 15130, 16385, 17425, 17690, 19045, 20165, 20450, 21905, 24650, 26245, 28225, 29585, 29930, 30277, 31330, 33125, 33490

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

			a(1) = 325 = A299707(1)^2 + 1 = 18^2 + 1 is expressible in two ways:
  325 = 17^2 + 6^2 = 15^2 + 10^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A050796, A299707, A300161, A300162, A300163, A300164, A300165, A300166.

A300161 Numbers n such that n^2 + 1 can be expressed as j^2 + k^2, j > k > 1, in more ways than for any smaller n.

Original entry on oeis.org

8, 18, 47, 132, 242, 268, 993, 1568, 3957, 5257, 14318, 18543, 43932, 66347, 72662, 161832, 330182, 1413443, 1732593, 2298668, 3315268, 9548768, 15926318, 24310918, 27028568, 51853693, 162166243, 420024818, 472936732, 599832943, 1892369318

Offset: 1

Hugo Pfoertner, Feb 27 2018

			a(1) = 8: 8^2 + 1 = 65 = 7^2 + 4^2,
a(2) = 18: 18^2 + 1 = 325 = 17^2 + 6^2 = 15^2 + 10^2,
a(3) = 47: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2,
a(4) = 132: 132^2 + 1 = 17425 = 129^2 + 28^2 = 127^2 + 36^2 = 120^2 + 55^2 = 116^2 + 63^2 = 105^2 + 80^2.

Cf. A050796, A299707, A299708, A300162, A300163.

a(17) from Hugo Pfoertner, Mar 08 2018
a(18)-a(21) from Robert Price, Mar 10 2018
a(22)-a(31) from Giovanni Resta, Mar 13 2018

A300163 Records in the number of ways to express a number of the form n^2 + 1 as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 15, 17, 23, 31, 35, 39, 47, 63, 71, 95, 127, 143, 161, 191, 215, 255, 287, 319, 383, 575, 639, 767, 959, 1151

Offset: 1

Hugo Pfoertner, Feb 27 2018

			a(6) = 8 because A300162(6) = A300161(6)^2 + 1 = 71825 is the smallest number expressible in 8 ways: 71825 = 265^2 + 40^2 = 260^2 + 65^2 = 257^2 + 76^2 = 247^2 + 104^2 = 236^2 + 127^2 = 215^2 + 160^2 = 208^2 + 169^2 = 191^2 + 188^2.

Cf. A060306, A071385, A299707, A299708, A300161, A300162, A300168.

a(17) from Hugo Pfoertner, Mar 08 2018
a(18)-a(21) from Robert Price, Mar 10 2018
a(22)-a(31) from Giovanni Resta, Mar 13 2018

A300168 Numbers of the form n^2+1 that can be expressed as j^2+k^2, j>k>1, gcd(j,k)=1, in more ways than any smaller number of this form.

Original entry on oeis.org

65, 2210, 58565, 4678570, 442765765, 5279766245, 2419804247185, 271780381692170, 28579081466859170, 4069607103295265285

Offset: 1

Hugo Pfoertner, Feb 27 2018

All ten known terms are squarefree. - Ray Chandler, Dec 31 2019
a(11) <= 1035219700200622531985 which is squarefree and expressible in 2047 ways. - Ray Chandler, Dec 24 2019
a(12) <= 4431331071224333359263505 which is squarefree and expressible in 4095 ways. - Ray Chandler, Dec 25 2019

			a(1) = 65 = 8^2 + 1 is the smallest expressible number (65 = 7^2 + 4^2),
a(2) = 2210 is expressible in 3 ways (2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2),
a(3) = 58565 is expressible in 7 ways,
a(4) = 4678570 is expressible in 15 ways,
a(5) = 442765765 is expressible in 31 ways.
Would a(6) be expressible in 63 ways?
Yes, a(6) = 5279766245 is expressible in 63 ways. - _Hugo Pfoertner_, Mar 02 2018
a(7) = 2419804247185 can be expressed in 127 ways.  This continues the progression that a(n) can be expressed in n^2-1 ways. - _Robert Price_, Mar 11 2018, updated by _Ray Chandler_, Dec 23 2019
a(8) = 271780381692170 can be expressed in 255 ways.
a(9) = 28579081466859170 can be expressed in 511 ways.
a(10) = 4069607103295265285 can be expressed in 1023 ways.

Cf. A300162, A300166, A300167.

a(7) from Robert Price, Mar 11 2018
a(7) corrected, a(8)-a(9) added by Ray Chandler, Dec 23 2019
a(10) added by Ray Chandler, Dec 31 2019

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

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A299707 Numbers m such that m^2 + 1 can be expressed in more than one way as j^2 + k^2 with j > k > 1.

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A299708 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1.

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A300161 Numbers n such that n^2 + 1 can be expressed as j^2 + k^2, j > k > 1, in more ways than for any smaller n.

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A300163 Records in the number of ways to express a number of the form n^2 + 1 as j^2 + k^2 with j > k > 1.

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A300168 Numbers of the form n^2+1 that can be expressed as j^2+k^2, j>k>1, gcd(j,k)=1, in more ways than any smaller number of this form.

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