OFFSET
1,2
COMMENTS
These are also the integer radii of circles around the origin that contain record numbers of lattice points. See A071383 for radii that are not necessarily integer. - Günter Rote, Sep 14 2023
REFERENCES
R. M. Sternheimer, Additional Remarks Concerning The Pythagorean Triplets, Journal of Recreational Mathematics, Vol. 30, No. 1, pp. 45-48, 1999-2000, Baywood NY.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..307
EXAMPLE
From Petros Hadjicostas, Jul 21 2019: (Start)
Squares 1^2, 2^2, 3^2, and 4^2 have 0 representations as the sum of two positive squares. (Thus, A088111(1) = 0 for the number of representations of 1^2.) Thus a(1) = 1.
Square 5^2 can be written as 3^2 + 4^2 only (here A088111(2) = 1). Thus, a(2) = 5.
Looking at sequence A046080, we see that for 5 <= n <= 24, only n^2 = 5^2, 10^2, 13^2, 15^2, 17^2, 20^2 can be written as a sum of two positive squares (in a single way) because 5^2 = 3^2 + 4^2, 10^2 = 6^2 + 8^2, 13^2 = 5^2 + 12^2, 17^2 = 8^2 + 15^2, and 20^2 = 12^2 + 16^2.
Since A046080(25) = 2 and A088111(3) = 2, we have that 25^2 can be written as a sum of two positive squares in two ways. Indeed, 25^2 = 7^2 + 24^2 = 15^2 + 20^2. Thus, a(3) = 25.
For 26 <= n <= 64, we see from sequence A046080 that n^2 cannot be written in more than 2 ways as a sum of two positive squares.
Since A046080(65) = 4, we see that 65^2 can be written as the sum of two positive squares in 4 ways. Indeed, 65^2 = 16^2 + 63^2 = 25^2 + 60^2 = 33^2 + 56^2 = 39^2 + 52^2. Thus, a(4) = 65.
(End)
PROG
(Python)
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=0, -1
for radius, expo in generate_A054994():
num_pyt = (prod((2*e+1) for e in expo)-1)//2
if num_pyt>record:
record = num_pyt
n += 1
print(radius, end="") # or record, for A088111
if n==26: break # stop after 26 entries
print(end=", ")
print() # Günter Rote, Sep 13 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Lekraj Beedassy, Dec 01 2003
EXTENSIONS
Corrected and extended by Ray Chandler, Jan 12 2012
Name edited by Petros Hadjicostas, Jul 21 2019
STATUS
approved