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Prime numbers that have an even number of monotone Bacher representations (A368276).
1

%I #22 Jan 02 2024 08:40:46

%S 5,7,13,17,23,43,53,59,61,71,79,83,107,109,113,127,131,137,139,167,

%T 181,191,193,199,211,223,227,239,241,257,271,277,293,307,313,317,331,

%U 337,347,353,359,367,379,389,401,421,431,439,449,457,461,467,479,499

%N Prime numbers that have an even 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 = 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 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]],EvenQ[A368276[#]]&] (* _Paolo Xausa_, Jan 02 2024 *)

%o (Julia)

%o using Nemo

%o println([n for n in 1:500 if iseven(A368276(n)) && is_prime(n)])

%o (Python)

%o from itertools import takewhile, islice

%o from sympy import nextprime, divisors

%o def A368277_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^1:

%o yield p

%o p = nextprime(p)

%o A368277_list = list(islice(A368277_gen(),30)) # _Chai Wah Wu_, Dec 19 2023

%Y Cf. A368276, A368278, A368207.

%K nonn

%O 1,1

%A _Peter Luschny_, Dec 19 2023