A368278 - OEIS

%I #25 Jan 02 2024 08:40:57

%S 2,3,11,19,29,31,37,41,47,67,73,89,97,101,103,149,151,157,163,173,179,

%T 197,229,233,251,263,269,281,283,311,349,373,383,397,409,419,433,443,

%U 463,487,491,521,523,557,577,587,601,607,619,659,661,673,677,701,719

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

%C 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 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 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 t[n_]:=t[n]=Select[Divisors[n],#^2<=n&];

%t A368276[n_]:=Total[t[n]]+Sum[Boole[wx<d*dx],{wx,Floor[n/2]},{dx,t[wx]},{d,t[n-wx]}];

%t Select[Prime[Range[200]],OddQ[A368276[#]]&] (* _Paolo Xausa_, Jan 02 2024 *)

%o (Julia)

%o using Nemo

%o println([n for n in 1:720 if isodd(A368276(n)) && is_prime(n)])

%o (Python)

%o from itertools import takewhile, islice

%o from sympy import divisors, nextprime

%o def A368278_gen(startvalue=2): # generator of terms >= startvalue

%o     p = max(nextprime(startvalue-1),2)

%o     while True:

%o         c = sum(takewhile(lambda x:x**2<=p,divisors(p))) & 1

%o         for wx in range(1,(p>>1)+1):

%o             for d1 in divisors(wx):

%o                 if d1**2 > wx:

%o                     break

%o                 m = p-wx

%o                 c = c+sum(1 for d in takewhile(lambda x:x**2<=m,divisors(m)) if wx<d*d1)&1

%o         if c:

%o             yield p

%o         p = nextprime(p)

%o A368278_list = list(islice(A368278_gen(),30)) # _Chai Wah Wu_, Dec 19 2023

%Y Cf. A368276, A368277, A368207.

%K nonn

%O 1,1

%A _Peter Luschny_, Dec 19 2023