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A368278 Prime numbers that have an odd number of monotone Bacher representations (A368276).

%I A368278 #29 Apr 26 2025 01:30:03
%S A368278 2,3,11,19,29,31,37,41,47,67,73,89,97,101,103,149,151,157,163,173,179,
%T A368278 197,229,233,251,263,269,281,283,311,349,373,383,397,409,419,433,443,
%U A368278 463,487,491,521,523,557,577,587,601,607,619,659,661,673,677,701,719
%N A368278 Prime numbers that have an odd number of monotone Bacher representations (A368276).
%C A368278 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 A368278 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 A368278 For n = 19, the 5 solutions are (w, x, y, z) = (0, 0, 1, 19), (1, 1, 2, 9), (1, 1, 3, 6), (1, 3, 4, 4), (2, 2, 3, 5).
%t A368278 t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];
%t A368278 A368276[n_]:=Total[t[n]]+Sum[Boole[wx<d*dx],{wx,Floor[n/2]},{dx,t[wx]},{d,t[n-wx]}];
%t A368278 Select[Prime[Range[200]],OddQ[A368276[#]]&] (* _Paolo Xausa_, Jan 02 2024 *)
%o A368278 (Julia)
%o A368278 using Nemo
%o A368278 println([n for n in 1:720 if isodd(A368276(n)) && is_prime(n)])
%o A368278 (Python)
%o A368278 from itertools import takewhile, islice
%o A368278 from sympy import divisors, nextprime
%o A368278 def A368278_gen(startvalue=2): # generator of terms >= startvalue
%o A368278     p = max(nextprime(startvalue-1),2)
%o A368278     while True:
%o A368278         c = sum(takewhile(lambda x:x**2<=p,divisors(p))) & 1
%o A368278         for wx in range(1,(p>>1)+1):
%o A368278             for d1 in divisors(wx):
%o A368278                 if d1**2 > wx:
%o A368278                     break
%o A368278                 m = p-wx
%o A368278                 c = c+sum(1 for d in takewhile(lambda x:x**2<=m,divisors(m)) if wx<d*d1)&1
%o A368278         if c:
%o A368278             yield p
%o A368278         p = nextprime(p)
%o A368278 A368278_list = list(islice(A368278_gen(),30)) # _Chai Wah Wu_, Dec 19 2023
%Y A368278 Cf. A368276, A368277, A368207.
%K A368278 nonn
%O A368278 1,1
%A A368278 _Peter Luschny_, Dec 19 2023