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A368277 Prime numbers that have an even number of monotone Bacher representations (A368276).

%I A368277 #26 Apr 26 2025 01:30:07
%S A368277 5,7,13,17,23,43,53,59,61,71,79,83,107,109,113,127,131,137,139,167,
%T A368277 181,191,193,199,211,223,227,239,241,257,271,277,293,307,313,317,331,
%U A368277 337,347,353,359,367,379,389,401,421,431,439,449,457,461,467,479,499
%N A368277 Prime numbers that have an even number of monotone Bacher representations (A368276).
%C A368277 We call a quadruple (w, x, y, z) of nonnegative integers a monotone Bacher representation of n if and only if n = w*x + y*z and w <= x < y <= z.
%H A368277 Roland Bacher, <a href="https://doi.org/10.1080/00029890.2023.2242034">A quixotic proof of Fermat's two squares theorem for prime numbers</a>, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; <a href="https://arxiv.org/abs/2210.07657">arXiv version</a>, arXiv:2210.07657 [math.NT], 2022.
%e A368277 For n = 13, the 4 solutions are (w, x, y, z) = (0, 0, 1, 13), (1, 1, 2, 6), (1, 1, 3, 4), (2, 2, 3, 3).
%t A368277 t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];
%t A368277 A368276[n_]:=Total[t[n]]+Sum[Boole[wx<d*dx],{wx,Floor[n/2]},{dx,t[wx]},{d,t[n-wx]}];
%t A368277 Select[Prime[Range[200]],EvenQ[A368276[#]]&] (* _Paolo Xausa_, Jan 02 2024 *)
%o A368277 (Julia)
%o A368277 using Nemo
%o A368277 println([n for n in 1:500 if iseven(A368276(n)) && is_prime(n)])
%o A368277 (Python)
%o A368277 from itertools import takewhile, islice
%o A368277 from sympy import nextprime, divisors
%o A368277 def A368277_gen(startvalue=2): # generator of terms >= startvalue
%o A368277     p = max(nextprime(startvalue-1),2)
%o A368277     while True:
%o A368277         c = sum(takewhile(lambda x:x**2<=p,divisors(p))) &1
%o A368277         for wx in range(1,(p>>1)+1):
%o A368277             for d1 in divisors(wx):
%o A368277                 if d1**2 > wx:
%o A368277                     break
%o A368277                 m = p-wx
%o A368277                 c = c+sum(1 for d in takewhile(lambda x:x**2<=m,divisors(m)) if wx<d*d1)&1
%o A368277         if c^1:
%o A368277             yield p
%o A368277         p = nextprime(p)
%o A368277 A368277_list = list(islice(A368277_gen(),30)) # _Chai Wah Wu_, Dec 19 2023
%Y A368277 Cf. A368276, A368278, A368207.
%K A368277 nonn
%O A368277 1,1
%A A368277 _Peter Luschny_, Dec 19 2023