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%I #46 Sep 15 2023 07:53:30
%S 1,5,25,65,325,1105,5525,27625,32045,160225,801125,1185665,5928325,
%T 29641625,48612265,243061325,1215306625,2576450045,12882250225,
%U 64411251125,157163452745,785817263725,3929086318625,10215624428425,11472932050385,51078122142125
%N 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.
%C 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
%D R. M. Sternheimer, Additional Remarks Concerning The Pythagorean Triplets, Journal of Recreational Mathematics, Vol. 30, No. 1, pp. 45-48, 1999-2000, Baywood NY.
%H Ray Chandler, <a href="/A088959/b088959.txt">Table of n, a(n) for n = 1..307</a>
%e From _Petros Hadjicostas_, Jul 21 2019: (Start)
%e 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.
%e Square 5^2 can be written as 3^2 + 4^2 only (here A088111(2) = 1). Thus, a(2) = 5.
%e 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.
%e 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.
%e 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.
%e 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.
%e (End)
%o (Python)
%o from math import prod
%o from sympy import isprime
%o primes_congruent_1_mod_4 = [5]
%o def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
%o while i>=len(primes_congruent_1_mod_4): # generate primes on demand
%o n = primes_congruent_1_mod_4[-1]+4
%o while not isprime(n): n += 4
%o primes_congruent_1_mod_4.append(n)
%o return primes_congruent_1_mod_4[i]
%o def generate_A054994():
%o TO_DO = {(1,())}
%o while True:
%o radius, exponents = min(TO_DO)
%o yield radius, exponents
%o TO_DO.remove((radius, exponents))
%o TO_DO.update(successors(radius,exponents))
%o def successors(r,exponents):
%o for i,e in enumerate(exponents):
%o if i==0 or exponents[i-1]>e:
%o yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
%o if exponents==() or exponents[-1]>0:
%o yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
%o n,record=0,-1
%o for radius,expo in generate_A054994():
%o num_pyt = (prod((2*e+1) for e in expo)-1)//2
%o if num_pyt>record:
%o record = num_pyt
%o n += 1
%o print(radius, end="") # or record, for A088111
%o if n==26: break # stop after 26 entries
%o print(end=", ")
%o print() # _Günter Rote_, Sep 13 2023
%Y Cf. A052199. Subsequence of A054994. Number of ways: see A088111. Where records occur in A046080.
%K nonn
%O 1,2
%A _Lekraj Beedassy_, Dec 01 2003
%E Corrected and extended by _Ray Chandler_, Jan 12 2012
%E Name edited by _Petros Hadjicostas_, Jul 21 2019
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