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

2, 3, 11, 19, 29, 31, 37, 41, 47, 67, 73, 89, 97, 101, 103, 149, 151, 157, 163, 173, 179, 197, 229, 233, 251, 263, 269, 281, 283, 311, 349, 373, 383, 397, 409, 419, 433, 443, 463, 487, 491, 521, 523, 557, 577, 587, 601, 607, 619, 659, 661, 673, 677, 701, 719

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

Table of n, a(n) for n=1..55.

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)
A368278_list = list(islice(A368278_gen(), 30)) # Chai Wah Wu, Dec 19 2023

Crossrefs

Cf. A368276, A368277, A368207.

Sequence in context: A306395 A363497 A135206 * A235629 A111802 A213894

Adjacent sequences: A368275 A368276 A368277 * A368279 A368280 A368281

Keyword

nonn

Author

Peter Luschny, Dec 19 2023

Status

approved