OFFSET
1,1
COMMENTS
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.
LINKS
Roland Bacher, A quixotic proof of Fermat's two squares theorem for prime numbers, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; arXiv version, arXiv:2210.07657 [math.NT], 2022.
EXAMPLE
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).
MATHEMATICA
t[n_]:=t[n]=Select[Divisors[n], #^2<=n&];
A368276[n_]:=Total[t[n]]+Sum[Boole[wx<d*dx], {wx, Floor[n/2]}, {dx, t[wx]}, {d, t[n-wx]}];
Select[Prime[Range[200]], OddQ[A368276[#]]&] (* Paolo Xausa, Jan 02 2024 *)
PROG
(Julia)
using Nemo
println([n for n in 1:720 if isodd(A368276(n)) && is_prime(n)])
(Python)
from itertools import takewhile, islice
from sympy import divisors, nextprime
def A368278_gen(startvalue=2): # generator of terms >= startvalue
p = max(nextprime(startvalue-1), 2)
while True:
c = sum(takewhile(lambda x:x**2<=p, divisors(p))) & 1
for wx in range(1, (p>>1)+1):
for d1 in divisors(wx):
if d1**2 > wx:
break
m = p-wx
c = c+sum(1 for d in takewhile(lambda x:x**2<=m, divisors(m)) if wx<d*d1)&1
if c:
yield p
p = nextprime(p)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Dec 19 2023
STATUS
approved