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

%I A071383 #63 Sep 15 2023 07:53:11
%S A071383 0,1,5,25,65,325,1105,4225,5525,27625,71825,138125,160225,801125,
%T A071383 2082925,4005625,5928325,29641625,77068225,148208125,243061325,
%U A071383 1215306625,3159797225,6076533125,12882250225,53716552825,64411251125
%N A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.
%C A071383 The number of lattice points (i,j) on the circle with i^2 + j^2 = a(n) is given by A071385(n).
%C A071383 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.
%C A071383 Also, numbers with a record number of divisors all of whose prime factors are of the form 4k + 1. - _Amiram Eldar_, Sep 12 2019
%C A071383 Indices of records of A004018. Apart from the first term, also indices of records of A002654. - _Jianing Song_, May 20 2021
%H A071383 Ray Chandler, <a href="/A071383/b071383.txt">Table of n, a(n) for n = 1..425</a> (first 97 terms from Ray Chandler, terms 98-365 from Amiram Eldar)
%H A071383 James Buddenhagen, <a href="https://groups.google.com/group/sci.math/msg/f2087083936610d3">Circle with 3 lattice points, thread in sci.math</a> (May 2002)
%H A071383 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a071383.pdf">Construction of the sequences A071383, A071384, A071385</a>
%F A071383 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
%F A071383 lim n ->infinity Log(a(n))/n = 1. [Conjectured by _Benoit Cloitre_, proved by _J. H. Conway_]
%o A071383 (PARI) 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
%o A071383 (Python)
%o A071383 from math import prod
%o A071383 from sympy import isprime
%o A071383 primes_congruent_1_mod_4 = [5]
%o A071383 def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
%o A071383     while i>=len(primes_congruent_1_mod_4): # generate primes on demand
%o A071383         n = primes_congruent_1_mod_4[-1]+4
%o A071383         while not isprime(n): n += 4
%o A071383         primes_congruent_1_mod_4.append(n)
%o A071383     return primes_congruent_1_mod_4[i]
%o A071383 def generate_A054994():
%o A071383     TO_DO = {(1,())}
%o A071383     while True:
%o A071383         radius, exponents = min(TO_DO)
%o A071383         yield radius, exponents
%o A071383         TO_DO.remove((radius, exponents))
%o A071383         TO_DO.update(successors(radius,exponents))
%o A071383 def successors(r,exponents):
%o A071383     for i,e in enumerate(exponents):
%o A071383         if i==0 or exponents[i-1]>e:
%o A071383             yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
%o A071383     if exponents==() or exponents[-1]>0:
%o A071383         yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
%o A071383 n,record,radius=1,1,0
%o A071383 print(radius, end="") # or record, for A071385
%o A071383 for radius,expo in generate_A054994():
%o A071383     num_points = 4*prod((e+1) for e in expo)
%o A071383     if num_points>record:
%o A071383         record = num_points
%o A071383         n += 1
%o A071383         print (",", radius, end="") # or record, for A071385
%o A071383         if n==27: break
%o A071383 print()
%o A071383 # _Günter Rote_, Sep 12 2023
%Y A071383 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.
%Y A071383 Indices of records of Sum_{d|n} kronecker(m, d): A230655 (m=-3), this sequence (m=-4), A279541 (m=-6).
%K A071383 nonn
%O A071383 1,3
%A A071383 _Hugo Pfoertner_, May 23 2002