A368276 Number of nonnegative representations of n = w*x + y*z with max(w, x) < min(y, z) and w <= x <= y <= z.

1, 1, 1, 3, 2, 3, 2, 3, 5, 4, 3, 6, 4, 4, 5, 9, 4, 7, 5, 9, 6, 7, 4, 11, 11, 7, 7, 11, 7, 13, 7, 10, 9, 10, 9, 19, 9, 9, 10, 17, 9, 17, 8, 14, 14, 13, 7, 21, 17, 14, 13, 17, 10, 20, 13, 22, 14, 15, 10, 26, 14, 13, 18, 28, 15, 22, 13, 19, 17, 25, 12, 33, 15, 18

Offset

1,4

Comments

Number of monotone Bacher representations (A368207) of n. 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..74.

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], 2023.

Example

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).

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]}];
Array[A368276, 100] (* Paolo Xausa, Jan 02 2024 *)

Prog

(Python)
from itertools import takewhile
from sympy import divisors
def A368276(n):
    c = sum(takewhile(lambda x: x**2 <= n, divisors(n)))
    for wx in range(1, (n >> 1) + 1):
        for d1 in divisors(wx):
            if d1**2 > wx:
                break
            m = n - wx
            c += sum(1
                for d in takewhile(lambda x: x**2 <= m, divisors(n - wx))
                if wx < d * d1)
    return c  # Chai Wah Wu, Dec 19 2023
(Julia)
using Nemo
function A368276(n)
    t(n) = (d for d in divisors(n) if d * d <= n)
    sum(sum(sum(1 for d in t(n - wx) if wx < d * dx; init=0)
            for dx in t(wx)) for wx in 1:div(n, 2); init=sum(t(n)))
end
println([A368276(n) for n in 1:74])  # Peter Luschny, Dec 19 2023

Crossrefs

Cf. A368207, A368276, A368457, A368458, A368459, A368580, A368581.

Sequence in context: A217618 A018837 A097618 * A039639 A023509 A115041

Adjacent sequences: A368273 A368274 A368275 * A368277 A368278 A368279

Keyword

nonn

Author

Peter Luschny, Dec 19 2023

Status

approved