A088959 Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of two positive squares in more ways than for any smaller number.

1, 5, 25, 65, 325, 1105, 5525, 27625, 32045, 160225, 801125, 1185665, 5928325, 29641625, 48612265, 243061325, 1215306625, 2576450045, 12882250225, 64411251125, 157163452745, 785817263725, 3929086318625, 10215624428425, 11472932050385, 51078122142125

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

Cf. A052199. Subsequence of A054994. Number of ways: see A088111. Where records occur in A046080.

Sequence in context: A108403 A007058 A071383 * A018782 A340388 A146665

Adjacent sequences: A088956 A088957 A088958 * A088960 A088961 A088962

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